MAT 1420 Precalculus Final Exam | Complete Practice
Questions & Detailed Solutions
Question 1
5
What is the domain of the function 𝑓(𝑥 ) = ?
√𝑥−3
• A. (3, ∞)
• B. [3, ∞)
• C. (−∞, 3)
• D. [3,4]
• Correct Answer: A
• Detailed Rationale: For the function to be defined, the expression
inside the square root must be strictly greater than zero because it
is in the denominator (𝑥 − 3 > 0). Solving this gives 𝑥 > 3, which
in interval notation is (3, ∞).
Question 2
Determine whether the function 𝑓(𝑥 ) = 𝑥 3 − 4𝑥is even, odd, or
neither.
• A. Even
• B. Odd
• C. Neither even nor odd
• D. Both even and odd
• Correct Answer: B
, • Detailed Rationale: A function is odd if 𝑓(−𝑥 ) = −𝑓 (𝑥 ). Testing
−𝑥: 𝑓 (−𝑥 ) = (−𝑥 )3 − 4(−𝑥 ) = −𝑥 3 + 4𝑥 = −(𝑥 3 − 4𝑥 ) =
−𝑓(𝑥 ). Thus, the function is odd.
Question 3
If 𝑓 (𝑥 ) = 2𝑥 2 − 3, what is the effect on the graph of 𝑓(𝑥 )if it is
transformed to 𝑔(𝑥 ) = 2(𝑥 − 1)2 − 3 + 4?
• A. Shifted right 1 unit and up 4 units
• B. Shifted left 1 unit and up 4 units
• C. Shifted right 1 unit and down 4 units
• D. Shifted left 1 unit and down 4 units
• Correct Answer: A
• Detailed Rationale: Replacing 𝑥with (𝑥 − 1)shifts the graph
horizontally to the right by 1unit. Adding 4to the entire function
shifts the graph vertically upward by 4units.
Question 4
𝑓(𝑥+ℎ)−𝑓(𝑥)
What is the difference quotient for the linear function
ℎ
𝑓(𝑥 ) = 3𝑥 − 2?
• A. 3
• B. 3ℎ
2
• C. 3 +
ℎ
• D. 0
• Correct Answer: A
, • Detailed Rationale: Evaluate 𝑓(𝑥 + ℎ) = 3(𝑥 + ℎ) − 2 = 3𝑥 +
3ℎ − 2. Then subtract 𝑓(𝑥 ): (3𝑥 + 3ℎ − 2) − (3𝑥 − 2) = 3ℎ.
3ℎ
Dividing by ℎyields = 3.
ℎ
Question 5
Given 𝑓(𝑥 ) = 2𝑥 + 1and 𝑔(𝑥 ) = 𝑥 2 − 3, what is the composite
function (𝑓 ∘ 𝑔)(2)?
• A. 3
• B. 5
• C. 9
• D. 11
• Correct Answer: B
• Detailed Rationale: First, evaluate 𝑔(2) = 22 − 3 = 4 − 3 = 1.
Next, substitute this into 𝑓: 𝑓 (1) = 2(1) + 1 = 3. Wait, let's
recalculate: 𝑔(2) = 4 − 3 = 1, then 𝑓(1) = 2(1) + 1 = 3. Let's
re-verify options. If options are A. 3, B. 5... wait, 𝑔(2) = 1, 𝑓(1) =
3. Let's check Option A. Yes, correct answer is A. Let's fix the letter
to A.
• Correct Answer: A
• Detailed Rationale: First, find 𝑔(2) = 22 − 3 = 1. Then, find
𝑓(𝑔(2)) = 𝑓 (1) = 2(1) + 1 = 3.
Question 6
𝑥
Find the inverse function 𝑓 −1 (𝑥 )for 𝑓 (𝑥 ) = for 𝑥 ≠ −1.
𝑥+1
𝑥
• A. 𝑓 −1 (𝑥 ) =
1−𝑥
, 𝑥+1
• B. 𝑓 −1 (𝑥 ) =
𝑥
1−𝑥
• C. 𝑓 −1 (𝑥 ) =
𝑥
𝑥
• D. 𝑓 −1 (𝑥 ) =
𝑥−1
• Correct Answer: A
𝑥 𝑦
• Detailed Rationale: Set 𝑦 = and swap 𝑥and 𝑦: 𝑥 = .
𝑥+1 𝑦+1
Multiply by (𝑦 + 1): 𝑥 (𝑦 + 1) = 𝑦 ⟹ 𝑥𝑦 + 𝑥 = 𝑦 ⟹ 𝑥 = 𝑦 −
𝑥
𝑥𝑦 ⟹ 𝑥 = 𝑦(1 − 𝑥 ) ⟹ 𝑦 = .
1−𝑥
Question 7
( ) 𝑥 2 + 1, 𝑥 < 2
Evaluate the piecewise function 𝑓 𝑥 = { at 𝑥 = 2.
3𝑥 − 2, 𝑥 ≥ 2
• A. 4
• B. 5
• C. 3
• D. 6
• Correct Answer: A
• Detailed Rationale: Since the condition for the second piece is
𝑥 ≥ 2, we use 𝑓 (𝑥 ) = 3𝑥 − 2when 𝑥 = 2. Substituting gives
3(2) − 2 = 6 − 2 = 4.
Question 8
What is the average rate of change of 𝑓 (𝑥 ) = 𝑥 3 over the interval [1,3]?
• A. 13
Questions & Detailed Solutions
Question 1
5
What is the domain of the function 𝑓(𝑥 ) = ?
√𝑥−3
• A. (3, ∞)
• B. [3, ∞)
• C. (−∞, 3)
• D. [3,4]
• Correct Answer: A
• Detailed Rationale: For the function to be defined, the expression
inside the square root must be strictly greater than zero because it
is in the denominator (𝑥 − 3 > 0). Solving this gives 𝑥 > 3, which
in interval notation is (3, ∞).
Question 2
Determine whether the function 𝑓(𝑥 ) = 𝑥 3 − 4𝑥is even, odd, or
neither.
• A. Even
• B. Odd
• C. Neither even nor odd
• D. Both even and odd
• Correct Answer: B
, • Detailed Rationale: A function is odd if 𝑓(−𝑥 ) = −𝑓 (𝑥 ). Testing
−𝑥: 𝑓 (−𝑥 ) = (−𝑥 )3 − 4(−𝑥 ) = −𝑥 3 + 4𝑥 = −(𝑥 3 − 4𝑥 ) =
−𝑓(𝑥 ). Thus, the function is odd.
Question 3
If 𝑓 (𝑥 ) = 2𝑥 2 − 3, what is the effect on the graph of 𝑓(𝑥 )if it is
transformed to 𝑔(𝑥 ) = 2(𝑥 − 1)2 − 3 + 4?
• A. Shifted right 1 unit and up 4 units
• B. Shifted left 1 unit and up 4 units
• C. Shifted right 1 unit and down 4 units
• D. Shifted left 1 unit and down 4 units
• Correct Answer: A
• Detailed Rationale: Replacing 𝑥with (𝑥 − 1)shifts the graph
horizontally to the right by 1unit. Adding 4to the entire function
shifts the graph vertically upward by 4units.
Question 4
𝑓(𝑥+ℎ)−𝑓(𝑥)
What is the difference quotient for the linear function
ℎ
𝑓(𝑥 ) = 3𝑥 − 2?
• A. 3
• B. 3ℎ
2
• C. 3 +
ℎ
• D. 0
• Correct Answer: A
, • Detailed Rationale: Evaluate 𝑓(𝑥 + ℎ) = 3(𝑥 + ℎ) − 2 = 3𝑥 +
3ℎ − 2. Then subtract 𝑓(𝑥 ): (3𝑥 + 3ℎ − 2) − (3𝑥 − 2) = 3ℎ.
3ℎ
Dividing by ℎyields = 3.
ℎ
Question 5
Given 𝑓(𝑥 ) = 2𝑥 + 1and 𝑔(𝑥 ) = 𝑥 2 − 3, what is the composite
function (𝑓 ∘ 𝑔)(2)?
• A. 3
• B. 5
• C. 9
• D. 11
• Correct Answer: B
• Detailed Rationale: First, evaluate 𝑔(2) = 22 − 3 = 4 − 3 = 1.
Next, substitute this into 𝑓: 𝑓 (1) = 2(1) + 1 = 3. Wait, let's
recalculate: 𝑔(2) = 4 − 3 = 1, then 𝑓(1) = 2(1) + 1 = 3. Let's
re-verify options. If options are A. 3, B. 5... wait, 𝑔(2) = 1, 𝑓(1) =
3. Let's check Option A. Yes, correct answer is A. Let's fix the letter
to A.
• Correct Answer: A
• Detailed Rationale: First, find 𝑔(2) = 22 − 3 = 1. Then, find
𝑓(𝑔(2)) = 𝑓 (1) = 2(1) + 1 = 3.
Question 6
𝑥
Find the inverse function 𝑓 −1 (𝑥 )for 𝑓 (𝑥 ) = for 𝑥 ≠ −1.
𝑥+1
𝑥
• A. 𝑓 −1 (𝑥 ) =
1−𝑥
, 𝑥+1
• B. 𝑓 −1 (𝑥 ) =
𝑥
1−𝑥
• C. 𝑓 −1 (𝑥 ) =
𝑥
𝑥
• D. 𝑓 −1 (𝑥 ) =
𝑥−1
• Correct Answer: A
𝑥 𝑦
• Detailed Rationale: Set 𝑦 = and swap 𝑥and 𝑦: 𝑥 = .
𝑥+1 𝑦+1
Multiply by (𝑦 + 1): 𝑥 (𝑦 + 1) = 𝑦 ⟹ 𝑥𝑦 + 𝑥 = 𝑦 ⟹ 𝑥 = 𝑦 −
𝑥
𝑥𝑦 ⟹ 𝑥 = 𝑦(1 − 𝑥 ) ⟹ 𝑦 = .
1−𝑥
Question 7
( ) 𝑥 2 + 1, 𝑥 < 2
Evaluate the piecewise function 𝑓 𝑥 = { at 𝑥 = 2.
3𝑥 − 2, 𝑥 ≥ 2
• A. 4
• B. 5
• C. 3
• D. 6
• Correct Answer: A
• Detailed Rationale: Since the condition for the second piece is
𝑥 ≥ 2, we use 𝑓 (𝑥 ) = 3𝑥 − 2when 𝑥 = 2. Substituting gives
3(2) − 2 = 6 − 2 = 4.
Question 8
What is the average rate of change of 𝑓 (𝑥 ) = 𝑥 3 over the interval [1,3]?
• A. 13
