Core Academic Skills Assessment Mathematics: Algebra and Functions — Questions and Answers
Question 1: If f(x) = 3x² − 2x + 1, what is f(−2)?
Correct answer: 17
f(−2) = 3(4) − 2(−2) + 1 = 12 + 4 + 1 = 17.
Question 2: Which value of x satisfies |2x − 5| = 9?
- x = 7 or x = −2 (Correct answer)
- x = 7 only
- x = −2 only
- x = 2 or x = −7
Correct answer: x = 7 or x = −2
2x − 5 = 9 gives x = 7; 2x − 5 = −9 gives x = −2.
Question 3: A function is defined as g(x) = (x + 3)/(x − 2). For which value of x is g(x) undefined?
- x = 2 (Correct answer)
- x = −3
- x = 0
- x = 3
Correct answer: x = 2
The function is undefined when the denominator equals zero: x − 2 = 0, so x = 2.
Question 4: What is the slope of the line passing through (−1, 4) and (3, −4)?
- −2 (Correct answer)
- 2
- −1/2
- 1/2
Correct answer: −2
Slope = (−4 − 4)/(3 − (−1)) = −8/4 = −2.
Question 5: If 5(x − 3) = 2x + 6, what is x?
Correct answer: 7
5x − 15 = 2x + 6 → 3x = 21 → x = 7.
Question 6: Which expression is equivalent to (2x³y²)(3xy⁴)?
- 6x⁴y⁶ (Correct answer)
- 5x⁴y⁶
- 6x³y⁶
- 6x⁴y⁸
Correct answer: 6x⁴y⁶
Multiply coefficients (2·3 = 6) and add exponents: x^(3+1) = x⁴, y^(2+4) = y⁶.
Question 7: The graph of y = −x² + 4 opens in which direction and has what y-intercept?
- Opens downward; y-intercept 4 (Correct answer)
- Opens upward; y-intercept 4
- Opens downward; y-intercept −4
- Opens upward; y-intercept −4
Correct answer: Opens downward; y-intercept 4
The negative leading coefficient means the parabola opens downward, and setting x = 0 gives y = 4.
Question 8: A line has equation 4x − 2y = 8. What is its slope and y-intercept?
- slope = 2, y-intercept = −4 (Correct answer)
- slope = −2, y-intercept = 4
- slope = 4, y-intercept = −8
- slope = 2, y-intercept = 4
Correct answer: slope = 2, y-intercept = −4
Rewrite as y = 2x − 4; slope is 2 and y-intercept is −4.
Question 9: Which of the following is a factor of x² − x − 12?
- (x − 4) (Correct answer)
- (x + 3) only
- (x − 3)
- (x + 12)
Correct answer: (x − 4)
x² − x − 12 = (x − 4)(x + 3), so (x − 4) is a factor.
Question 10: If h(x) = 2x + 1 and k(x) = x², what is k(h(3))?
Correct answer: 49
h(3) = 7, then k(7) = 7² = 49.
Question 11: Solve the inequality: −3x + 6 > 12.
- x < −2 (Correct answer)
- x > −2
- x < 2
- x > 6
Correct answer: x < −2
−3x > 6 → x < −2 (inequality flips when dividing by a negative).
Question 12: Which equation represents a line parallel to y = (2/3)x − 5?
- y = (2/3)x + 7 (Correct answer)
- y = (3/2)x − 5
- y = −(2/3)x + 1
- y = (2/3)x − 5
Correct answer: y = (2/3)x + 7
Parallel lines have the same slope (2/3) but different y-intercepts.
Question 13: What is the solution set of x² = 25?
- x = 5 or x = −5 (Correct answer)
- x = 5 only
- x = −5 only
- x = ±√5
Correct answer: x = 5 or x = −5
Taking the square root of both sides gives x = ±5.
Question 14: A car rental company charges $30 per day plus $0.15 per mile. Which function gives the total cost C for d days and m miles?
- C = 30d + 0.15m (Correct answer)
- C = 30m + 0.15d
- C = 30 + 0.15(d + m)
- C = 0.15d + 30m
Correct answer: C = 30d + 0.15m
The daily rate multiplies by days and the mileage rate multiplies by miles: C = 30d + 0.15m.
Question 15: Simplify: (x² − 9)/(x + 3) for x ≠ −3.
- x − 3 (Correct answer)
- x + 3
- x² − 3
- (x − 3)(x + 3)
Correct answer: x − 3
Factor the numerator: (x − 3)(x + 3)/(x + 3) = x − 3.
Question 16: Which of the following tables of values represents a linear function?
- x: 0,1,2,3 → y: 4,7,10,13 (Correct answer)
- x: 0,1,2,3 → y: 1,2,4,8
- x: 0,1,2,3 → y: 0,1,4,9
- x: 0,1,2,3 → y: 5,4,2,−1
Correct answer: x: 0,1,2,3 → y: 4,7,10,13
The differences between consecutive y-values are constant (3), confirming a linear function.
Question 17: If 2^x = 32, what is x?
Correct answer: 5
32 = 2⁵, so x = 5.
Question 18: Which system of equations has no solution?
- y = 2x + 1 and y = 2x − 3 (Correct answer)
- y = x + 2 and y = −x + 2
- y = 3x and y = x + 4
- y = −x + 1 and y = x − 1
Correct answer: y = 2x + 1 and y = 2x − 3
Two lines with the same slope (2) but different y-intercepts are parallel and never intersect.
Question 19: What is the range of f(x) = x² + 3?
- y ≥ 3 (Correct answer)
- All real numbers
- y ≤ 3
- y > 0
Correct answer: y ≥ 3
Since x² ≥ 0 for all real x, the minimum value of f(x) is 3, so y ≥ 3.
Question 20: Solve for x: (x/4) + 3 = (x/2) − 1.
Correct answer: 16
Multiply through by 4: x + 12 = 2x − 4 → x = 16.
Question 21: The function y = 2(x − 1)² + 3 is in vertex form. What is its vertex?
- (1, 3) (Correct answer)
- (−1, 3)
- (1, −3)
- (2, 3)
Correct answer: (1, 3)
Vertex form y = a(x − h)² + k has vertex (h, k); here h = 1 and k = 3.
