Replace x with 5. Remember that 3x means 3 times x.
Step-by-step solution
Function notation f(5) means "the output when the input is 5."
Substitute 5 for x, using parentheses: f(5) = 3(5) + 4.
Multiply first: 3(5) = 15.
Then add: 15 + 4 = 19.
Answer: f(5) = 19
Check: The point (5, 19) should satisfy y = 3x + 4: 3 \cdot 5 + 4 = 19 ✓ Also, f(4) = 16 and f(6) = 22, and 19 sits exactly halfway between them, as it should for a line with slope 3.
Tip:3x means "3 timesx." Write 3(5) — never "35" — and multiply before you add. The parentheses habit pays off later, when the input is negative.
Question 2 of 36
No calculatorFill in
If g(x) = x^2 + 1, what is g(4)?
g(4) =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Replace x with 4. Square first, then add.
Step-by-step solution
g(4) means "the output of g when the input is 4."
Substitute: g(4) = (4)^2 + 1.
Exponent first: 4^2 = 4 \times 4 = 16.
Add: 16 + 1 = 17.
Answer: g(4) = 17
Check: Squaring is repeated multiplication, not doubling: 4^2 = 16, not 8. So 16 + 1 = 17 ✓
Tip: A common slip is 4^2 = 8 (multiplying by 2 instead of squaring), which would give 9. Squaring means multiplying the number by itself.
Question 3 of 36
No calculator
If f(x) = 10 - 2x, what is f(-3)?
Hint
Put -3 inside parentheses: 10 - 2(-3). What is a negative times a negative?
Step-by-step solution
Substitute -3 for x using parentheses: f(-3) = 10 - 2(-3).
Multiply: 2(-3) = -6.
Subtracting a negative is adding: 10 - (-6) = 10 + 6.
Add: 10 + 6 = 16.
Answer: f(-3) = 16
Check: The function decreases by 2 for each step right, so moving left 3 steps from f(0) = 10 should increase it by 2 \times 3 = 6: 10 + 6 = 16 ✓
Tip: Always wrap negative inputs in parentheses. "Minus a negative" becomes "plus."
Question 4 of 36
No calculator
A function is defined by the rule h(x) = 7x - 12. What output does h give for an input of 4?
Hint
Replace x with 4. The 7 is attached to x, so it multiplies.
Step-by-step solution
h(4) means "the output of h when the input is 4."
Substitute 4 for x, using parentheses: h(4) = 7(4) - 12.
Multiply first: 7(4) = 28.
Then subtract: 28 - 12 = 16.
Answer: h(4) = 16
Check: Work backwards: if the output is 16, then 7x = 16 + 12 = 28, so x = 28 \div 7 = 4 ✓
Tip: A number written right next to x always means multiplication. Write the parentheses 7(4) so you never read it as "74," and always multiply before you subtract.
Question 5 of 36
No calculatorFill in
Let f(x) = \dfrac{2}{3}x + 5. What is f(-9)?
f(-9) =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put -9 in parentheses. Two-thirds of a number: divide by 3, then multiply by 2.
Step-by-step solution
Substitute -9 for x in parentheses: f(-9) = \dfrac{2}{3}(-9) + 5.
Multiply the fraction by -9: divide by the denominator first, -9 \div 3 = -3, then multiply by the numerator, -3 \times 2 = -6.
Add 5: -6 + 5 = -1.
Answer: f(-9) = -1
Check:f(0) = 5 and the slope is \tfrac{2}{3}, so moving 9 units left lowers the output by \tfrac{2}{3} \times 9 = 6: 5 - 6 = -1 ✓
Tip: When the input is a multiple of the denominator, divide first; the numbers stay small. A positive times a negative is negative, so \tfrac{2}{3}(-9) must be negative before you add 5.
Question 6 of 36
No calculator
A gym charges a one-time sign-up fee plus a monthly fee. The total cost, in dollars, for m months of membership is
C(m) = 35m + 50
What is the total cost of an 8-month membership?
Hint
The input is the number of months. Find C(8), multiplying before adding.
Step-by-step solution
The input is the number of months, so m = 8.
Substitute: C(8) = 35(8) + 50.
Multiply: 35 \times 8 = 280. This is the monthly fees for 8 months.
Add the one-time sign-up fee: 280 + 50 = 330.
Answer: $330
Check: Estimate: about $35 a month for 8 months is a bit under $300; add the $50 fee and you get a little over $300. $330 fits ✓
Tip: In a linear cost function C(m) = (\text{rate})\,m + (\text{fee}), only the rate is multiplied by the input. The fee is paid once, so it is added once.
Question 7 of 36
No calculator
Use the line drawn below, which represents the function f.
Based on the graph, what is f(-2), the output of f when x = -2?
Hint
Find -2 on the horizontal axis. Move straight up or down to the line and read the y-value there.
Step-by-step solution
f(-2) is the output when the input is x = -2.
On the horizontal (x) axis, find -2 (two units left of the origin).
Move straight down until you reach the line. The line is at y = -3 there, so the point is (-2, -3).
The output is the y-coordinate: f(-2) = -3.
Answer: f(-2) = -3
Check: The line passes through (0, -2) and (4, 0), so its slope is \frac{0 - (-2)}{4 - 0} = \frac{1}{2} and f(x) = \frac{1}{2}x - 2. Then f(-2) = -1 - 2 = -3 ✓
Tip: The number inside the parentheses is always an x-value. Go across to it first, then up or down to the graph; never start on the y-axis.
Question 8 of 36
If f(x) = x^2 - 3x + 2, what is the value of f(-2)?
Hint
Substitute (-2) for bothx's. What is (-2)^2? What is -3(-2)?
Step-by-step solution
Substitute with parentheses: f(-2) = (-2)^2 - 3(-2) + 2.
Tip: Two sign traps live in this question: (-2)^2 is +4, and -3 \times -2 is +6. The wrong answers 4, 0 and -8 come from missing one or both.
Question 9 of 36
Fill in
A plumber charges according to the function
C(h) = 65h + 45
where C(h) is the total charge in dollars for a job that takes h hours. What is the total charge for a job that takes 3.5 hours?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Replace h with 3.5. Multiply before adding.
Step-by-step solution
The input is the number of hours: h = 3.5.
Substitute: C(3.5) = 65(3.5) + 45.
Multiply: 65 \times 3.5 = 227.5 (that is 65 \times 3 = 195 plus 65 \times 0.5 = 32.5).
Add the flat fee: 227.5 + 45 = 272.5.
Answer: $272.50
Check: Estimate: $65 per hour for about 3.5 hours is a bit under $230, plus the $45 call-out fee ≈ $275 ✓
Tip: In a linear cost function C(h) = mh + b, m is the hourly rate and b is the flat fee (C(0) = 45 — what you pay even for a zero-hour visit).
Question 10 of 36
A ball is tossed upward from a balcony. Its height, in feet, after t seconds is
h(t) = -16t^2 + 48t + 6
What is the height of the ball after 2 seconds?
Hint
Find h(2). Square the 2 first, then multiply by -16.
Step-by-step solution
Substitute t = 2: h(2) = -16(2)^2 + 48(2) + 6.
Exponent first: 2^2 = 4.
Multiply: -16(4) = -64 and 48(2) = 96.
Add: -64 + 96 + 6 = 38.
Answer: 38 feet
Check:h(1) = -16 + 48 + 6 = 38 too! The ball is at 38 ft at t = 1 (going up) and at t = 2 (coming down) — that symmetry is typical of a parabola, with the top at t = 1.5 ✓
Tip: In -16t^2, only t is squared and the result stays negative. Gravity always pulls the height down.
Question 11 of 36
Drop-down
Let f(x) = 2x^2 - 5. Complete the statements.
f(3) =
f(-3) =
Choose an answer from each menu.
Hint
Only x is squared — not the 2. And a negative number squared is positive.
Check:h(0) = 32 (the platform height) and h(2) = -64 + 32 + 32 = 0 (she reaches the water at 2 seconds). A height of 20 ft at 1.5 s fits between 32 ft and 0 ft as she falls ✓
Tip: In -16t^2, square onlyt, then multiply by -16. On the calculator, type (-)16×1.5x² so the square applies to 1.5 alone.
Question 15 of 36
No calculatorHot spot
Let k(x) = x^2 - 4.
Click the point on the coordinate plane that represents \big(-1,\ k(-1)\big).
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
First find k(-1). Remember that (-1)^2 is positive. Then plot (input, output).
Step-by-step solution
Evaluate: k(-1) = (-1)^2 - 4.
Exponent first: (-1)^2 = (-1)(-1) = 1.
Subtract: 1 - 4 = -3. So k(-1) = -3.
The point is (\text{input}, \text{output}) = (-1, -3).
Plot it: from the origin move 1 left and 3 down.
Answer: the point (-1, -3)
Check:k(1) = 1 - 4 = -3 too, so (1, -3) is also on the graph. A parabola x^2 + c is symmetric about the y-axis, so the outputs at -1 and 1 must match ✓
Tip: Squaring a negative number gives a positive result. If you wrote (-1)^2 = -1 you would land on (-1, -5), two units too low.
Question 16 of 36
No calculatorFill in
For the function g(x) = -5x + 8, exactly one input produces an output of 33. Find that input.
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Here 33 is the output. Set -5x + 8 equal to 33 and solve. Watch the sign when you divide.
Step-by-step solution
"g(x) = 33" gives the output; you need the input.
Write the equation: -5x + 8 = 33.
Subtract 8 from both sides: -5x = 25.
Divide both sides by -5: x = \dfrac{25}{-5} = -5.
Answer: x = -5
Check:g(-5) = -5(-5) + 8 = 25 + 8 = 33 ✓
Tip: The output 33 is larger than the starting value g(0) = 8, but the slope is negative (the function goes down as x goes right). So the input must be to the left of 0, a negative number. This quick sign check catches the common answer x = 5.
Question 17 of 36
No calculator
The table shows some inputs and outputs of the function f(x) = 2x^2 - x.
x
-2
-1
0
1
3
f(x)
10
3
0
1
?
Which number belongs in the empty cell?
Hint
Find f(3). Square the 3 first, then multiply by 2, then subtract 3.
Step-by-step solution
The empty cell is under x = 3, so you need f(3).
Substitute: f(3) = 2(3)^2 - 3.
Exponent first: 3^2 = 9.
Multiply: 2 \times 9 = 18.
Subtract: 18 - 3 = 15.
Answer: 15
Check: Test the rule on a cell you already know: f(-2) = 2(4) - (-2) = 8 + 2 = 10 ✓ matches the table, so the rule is being applied correctly.
Tip: Before filling a table, test your method on one cell that is already filled in. If you get the table's number, your method is right.
Question 18 of 36
No calculator
Jordan evaluated f(x) = -x^2 + 4x at x = -3. His work is shown.
Step 1:f(-3) = -(-3)^2 + 4(-3)
Step 2:= -(-9) + 4(-3)
Step 3:= 9 - 12
Step 4:= -3
Which statement is true?
Hint
Check each step separately. What is (-3)^2? Then what does the negative sign in front do?
Step-by-step solution
Step 1 substitutes -3 for both x's with parentheses. Correct.
Step 2 claims (-3)^2 = -9. But (-3)^2 = (-3)(-3) = +9. This is the first error.
Correct Step 2: -(9) + 4(-3) = -9 + 4(-3).
Correct Step 3: 4(-3) = -12, so -9 - 12.
Correct Step 4: -9 - 12 = -21.
Answer: The first error is in Step 2; the correct value is f(-3) = -21.
Tip: Undo operations in the reverse order they were done to x: square, times 2, plus 3 → minus 3, divide by 2, square root. And x^2 = 25 always has two answers.
Question 20 of 36
No calculatorDrop-down
Let f(x) = x^2 - 2x and g(x) = 4 - 3x. Complete the statements.
f(-2) =
g(-2) =
So f(-2) is g(-2).
Choose an answer from each menu.
Hint
Use parentheses for -2 in both functions. Two negatives multiplied give a positive.
Unused tiles: 5 comes from 1 - (-4), treating (-2)^2 as -4; and -5 comes from 1 - 6, treating 3^2 as 3 \times 2.
Answer: g(-2) = -3, g(0) = 1, g(3) = -8
Check:g(2) = 1 - 4 = -3, the same as g(-2): the rule only uses x^2, so opposite inputs must give equal outputs ✓
Tip: In 1 - x^2, the exponent is done before the subtraction. Square, then subtract from 1.
Question 22 of 36
A phone plan costs C(n) = 12 + 0.4n dollars per month, where n is the number of gigabytes of data used.
Priya says the cost for 50 gigabytes is $620, because 12 + 0.4 = 12.4 and 12.4 \times 50 = 620.
Which statement is true?
Hint
In 12 + 0.4n, which operation is done to n first: the adding or the multiplying?
Step-by-step solution
Substitute n = 50: C(50) = 12 + 0.4(50).
Order of operations: multiply before adding. 0.4 \times 50 = 20.
Add the base charge: 12 + 20 = 32.
Priya added 12 + 0.4first, which multiplies the $12 base fee by 50 as well. That is her error.
Answer: Priya added before multiplying; the correct cost is $32.
Check: $0.40 per gigabyte for 50 gigabytes is $20; add the $12 monthly base: $32. A $620 phone bill for this plan is clearly unreasonable ✓
Tip: In a function like 12 + 0.4n, only the 0.4 is attached to n. A quick reasonableness check (is the answer a sensible bill?) catches order-of-operations mistakes.
Question 23 of 36
Fill in
The function F(c) = 1.8c + 32 converts a temperature of c degrees Celsius to degrees Fahrenheit. A weather report from Canada gives a morning low of -15 degrees Celsius.
What is this temperature in degrees Fahrenheit?
°F
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find F(-15). Put -15 in parentheses and multiply before adding 32.
Step-by-step solution
The input is the Celsius temperature: c = -15.
Substitute: F(-15) = 1.8(-15) + 32.
Multiply: 1.8 \times 15 = 27, and a positive times a negative is negative, so 1.8(-15) = -27.
Add: -27 + 32 = 5.
Answer: 5 °F
Check:-15 °C is well below freezing (0 °C = 32 °F), so the Fahrenheit value should also be well below 32. 5 °F fits ✓
Tip: Real-world inputs can be negative. Keep the sign inside parentheses; the answer 59 comes from forgetting the negative (27 + 32).
Question 24 of 36
No calculatorFill in
A parabola representing the function g is drawn on the grid.
Using only the graph, find the sum g(-2) + g(2).
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Read the y-value of the curve at x = -2 and at x = 2 separately. Then add those two outputs.
Step-by-step solution
g(-2): go to x = -2 on the horizontal axis and move down to the curve. It is at y = -5, so g(-2) = -5.
g(2): go to x = 2 and move up to the curve. It is at y = 3, so g(2) = 3. (This point is the mirror image of the y-intercept (0, 3) across the vertex line x = 1.)
Tip: When the problem gives the result (a total bill), work backwards: undo the "+60" first, then the "×85". Plugging each answer choice into C(h) is a fast backup check on the GED.
Question 29 of 36
A bakery sells muffins. For n muffins made and sold in a day, its revenue and cost, in dollars, are
R(n) = 3.25n \qquad C(n) = 0.75n + 120
Profit is revenue minus cost. What is the bakery's profit on a day when it makes and sells 200 muffins?
Hint
Find R(200) and C(200) separately, then subtract: R(200) - C(200).
Step-by-step solution
Revenue: R(200) = 3.25(200) = 650.
Cost: C(200) = 0.75(200) + 120 = 150 + 120 = 270.
Profit: R(200) - C(200) = 650 - 270 = 380.
Answer: $380
Check: Each muffin brings in 3.25 - 0.75 = \$2.50 more than it costs to make, so 200 muffins give 2.50 \times 200 = 500. Subtract the $120 daily fixed cost: 500 - 120 = 380 ✓
Tip: Evaluate each function fully, including the fixed cost, before you subtract. Profit questions on the GED often hide the constant term in the cost function.
Question 30 of 36
No calculatorFill in
The table shows inputs and outputs of the function f(x) = 4x^2 - 2x.
x
-1
-\tfrac{1}{2}
0
\tfrac{1}{2}
f(x)
6
?
0
0
What is the missing output?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find f\left(-\tfrac{1}{2}\right). What is \left(-\tfrac{1}{2}\right)^2? What is -2 \times \left(-\tfrac{1}{2}\right)?
Step-by-step solution
Substitute with parentheses: f\left(-\tfrac{1}{2}\right) = 4\left(-\tfrac{1}{2}\right)^2 - 2\left(-\tfrac{1}{2}\right).
Exponent first: \left(-\tfrac{1}{2}\right)^2 = \tfrac{1}{4} (negative times negative is positive).
Check: Test the method on a known cell: f\left(\tfrac{1}{2}\right) = 4 \cdot \tfrac{1}{4} - 2 \cdot \tfrac{1}{2} = 1 - 1 = 0 ✓ matches the table.
Tip: Fractions and negatives together are where most slips happen. Write every step: square the fraction, multiply, then watch each sign. Dropping either one of the two signs gives the wrong answer 0; dropping both gives -2.
Question 31 of 36
No calculatorSelect all
Students evaluated the function q(x) = 3(x - 2)^2 at different inputs. Select all the evaluations that are correct.
Select all that apply.
Hint
For each one: subtract inside the parentheses, square that result, and only then multiply by 3.
Step-by-step solution
Order for q(x) = 3(x - 2)^2: parentheses → exponent → multiply by 3.
q(0) = 3(-2)^2 = 3(4) = 12. The work shown is correct ✓
q(5) = 3(3)^2 = 3(9) = 27, not 81. The 81 squares 3 \cdot 3 = 9. ✗
q(-1) = 3(-3)^2 = 3(9) = 27. Correct ✓
q(4) = 3(2)^2 = 3(4) = 12, not 36. The 36 squares 3 \cdot 2 = 6. ✗
q(1) = 3(-1)^2 = 3(1) = 3, not -3. A negative squared is positive. ✗
q(2) = 3(0)^2 = 3(0) = 0, not 3. Three times zero is zero. ✗
Answer: q(0) = 12 and q(-1) = 27 are the correct evaluations.
Check:q(0) = 12 and q(4) = 12 are equal, as they should be: 0 and 4 are both 2 units from x = 2, the vertex of this parabola ✓
Tip: The coefficient outside the parentheses is applied last. Squaring it by mistake is the most common error with this kind of function.
Question 32 of 36
Fill in
A worker on a roof accidentally drops a hammer. The hammer's height above the ground, in feet, t seconds after it is dropped is
h(t) = 144 - 16t^2
After how many seconds does the hammer hit the ground?
seconds
Type your answer in the box. Fractions like 3/4 are fine.
Hint
"Hits the ground" means the height is 0. Solve 144 - 16t^2 = 0. Can time be negative here?
Step-by-step solution
On the ground, the height is 0, so set h(t) = 0: 144 - 16t^2 = 0.
Add 16t^2 to both sides: 144 = 16t^2.
Divide by 16: t^2 = 9.
Square root: t = 3 or t = -3.
Time after the drop cannot be negative, so reject t = -3.
Answer: 3 seconds
Check:h(3) = 144 - 16(9) = 144 - 144 = 0 ✓ and h(0) = 144 ft is the roof height, which makes sense.
Tip: In word problems, solve completely and then ask whether each solution makes sense. Algebra gives \pm 3, but only the positive time fits the situation. Do not answer 9 — that is t^2, not t.
Question 33 of 36
No calculator
If f(x) = 5 - 2x, which expression is equal to f(x + 3)?
Hint
Replace everyx in the rule with the whole expression (x + 3), in parentheses. Then distribute.
Step-by-step solution
f(x + 3) means: use x + 3 as the input. Replace x with (x + 3): f(x + 3) = 5 - 2(x + 3).
Distribute -2 to both terms: -2 \cdot x = -2x and -2 \cdot 3 = -6.
Tip: You cannot find R(15) - R(10) by computing R(5). R(5) = 3500, which is not the difference — evaluate each function value separately.
Question 36 of 36
A food truck owner models the daily profit, in dollars, from selling x tacos as
P(x) = -0.05x^2 + 9x - 150
She compares a day when she sells 60 tacos with a day when she sells 120 tacos. Which statement is true?
Hint
Find P(60) and P(120) separately. Square the number of tacos before multiplying by -0.05.
Step-by-step solution
P(60) = -0.05(60)^2 + 9(60) - 150.
60^2 = 3600, and -0.05(3600) = -180.
9(60) = 540.
P(60) = -180 + 540 - 150 = 210.
P(120) = -0.05(120)^2 + 9(120) - 150.
120^2 = 14{,}400, and -0.05(14{,}400) = -720.
9(120) = 1080.
P(120) = -720 + 1080 - 150 = 210.
Both days give the same profit.
Answer: Both days earn the same profit, $210.
Check: A parabola is symmetric about its vertex. Here the vertex is at x = -\frac{9}{2(-0.05)} = 90, exactly halfway between 60 and 120, so equal outputs make sense ✓
Tip: Quadratic models are not proportional: doubling the input does not double the output. Selling more tacos can even lower profit once costs (the -0.05x^2 term) grow fast.
