Mathematics: Algebra and Functions Flashcards

If f(x) = 3x² − 2x + 1, what is f(−2)?

Answer: 17

f(−2) = 3(4) − 2(−2) + 1 = 12 + 4 + 1 = 17.

Which value of x satisfies |2x − 5| = 9?

Answer: x = 7 or x = −2

2x − 5 = 9 gives x = 7; 2x − 5 = −9 gives x = −2.

A function is defined as g(x) = (x + 3)/(x − 2). For which value of x is g(x) undefined?

Answer: x = 2

The function is undefined when the denominator equals zero: x − 2 = 0, so x = 2.

What is the slope of the line passing through (−1, 4) and (3, −4)?

Answer: −2

Slope = (−4 − 4)/(3 − (−1)) = −8/4 = −2.

If 5(x − 3) = 2x + 6, what is x?

Answer: 7

5x − 15 = 2x + 6 → 3x = 21 → x = 7.

Which expression is equivalent to (2x³y²)(3xy⁴)?

Answer: 6x⁴y⁶

Multiply coefficients (2·3 = 6) and add exponents: x^(3+1) = x⁴, y^(2+4) = y⁶.

The graph of y = −x² + 4 opens in which direction and has what y-intercept?

Answer: Opens downward; y-intercept 4

The negative leading coefficient means the parabola opens downward, and setting x = 0 gives y = 4.

A line has equation 4x − 2y = 8. What is its slope and y-intercept?

Answer: slope = 2, y-intercept = −4

Rewrite as y = 2x − 4; slope is 2 and y-intercept is −4.

Which of the following is a factor of x² − x − 12?

Answer: (x − 4)

x² − x − 12 = (x − 4)(x + 3), so (x − 4) is a factor.

If h(x) = 2x + 1 and k(x) = x², what is k(h(3))?

Answer: 49

h(3) = 7, then k(7) = 7² = 49.

Solve the inequality: −3x + 6 > 12.

Answer: x < −2

−3x > 6 → x < −2 (inequality flips when dividing by a negative).

Which equation represents a line parallel to y = (2/3)x − 5?

Answer: y = (2/3)x + 7

Parallel lines have the same slope (2/3) but different y-intercepts.

What is the solution set of x² = 25?

Answer: x = 5 or x = −5

Taking the square root of both sides gives x = ±5.

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?

Answer: C = 30d + 0.15m

The daily rate multiplies by days and the mileage rate multiplies by miles: C = 30d + 0.15m.

Simplify: (x² − 9)/(x + 3) for x ≠ −3.

Answer: x − 3

Factor the numerator: (x − 3)(x + 3)/(x + 3) = x − 3.

Which of the following tables of values represents a linear function?

Answer: x: 0,1,2,3 → y: 4,7,10,13

The differences between consecutive y-values are constant (3), confirming a linear function.

If 2^x = 32, what is x?

Answer: 5

32 = 2⁵, so x = 5.

Which system of equations has no solution?

Answer: y = 2x + 1 and y = 2x − 3

Two lines with the same slope (2) but different y-intercepts are parallel and never intersect.

What is the range of f(x) = x² + 3?

Answer: y ≥ 3

Since x² ≥ 0 for all real x, the minimum value of f(x) is 3, so y ≥ 3.

Solve for x: (x/4) + 3 = (x/2) − 1.

Answer: 16

Multiply through by 4: x + 12 = 2x − 4 → x = 16.