Further transformations: |f(x)|, f(|x|), 1/f(x), [f(x)]^2 practice questions.

Let f(x) = x² - 4. Determine which of the following points lies on the graph of y = |f(x)|.

• (1, -3)
• (-1, -3)
• (1, 3)
• (2, 4)

Answer: (1, 3)

f(1) = 1 - 4 = -3, and |f(x)| reflects negative parts of the graph in the x-axis, so the point (1, -3) becomes (1, 3).

Let f(x) = x - 2 and g(x) = f(|x|). Which statement about g is correct?

• g is even: symmetric about the y-axis
• g(x) ≥ 0 for all real x
• The point (-2, -4) lies on the graph of g
• g(x) = f(x) for all x < 0

Answer: g is even: symmetric about the y-axis

g(x) = |x| - 2 satisfies g(-x) = g(x), so it is even: the part of f for x ≥ 0 is kept and reflected in the y-axis. Note g(0) = -2 < 0 and g(-2) = 0, so the other statements fail.

Let f(x) = x² - 5x + 6. State the equations of the vertical asymptotes of the graph of y = 1/f(x).

• x = -2 and x = -3
• x = 2 and x = 3
• y = 2 and y = 3
• x = 5 and x = 6

Answer: x = 2 and x = 3

f(x) = (x - 2)(x - 3), and the zeros of f become vertical asymptotes of 1/f, so the asymptotes are x = 2 and x = 3.

A continuous function f has range [-2, 5]. Determine the range of y = |f(x)|.

• [2, 5]
• [-2, 5]
• [0, 2]
• [0, 5]

Answer: [0, 5]

Since f attains every value between -2 and 5, |f| attains every value from 0 (where f = 0) up to 5; values of f in [-2, 0) map to (0, 2], which is already covered.

The function f has zeros at x = -1 and x = 4 only. Determine the zeros of y = f(|x|).

• x = -1 and x = 4
• x = ±1 and x = ±4
• x = 4 and x = -4
• x = 1 and x = -4

Answer: x = 4 and x = -4

f(|x|) = 0 requires |x| = -1 (impossible) or |x| = 4, giving x = ±4. Zeros of f at negative x-values are discarded because |x| ≥ 0.

Let f(x) = cos x. Determine the period of the graph of y = [f(x)]².

• 2π
• π
• π/2
• 4π

Answer: π

cos²x = (1 + cos 2x)/2, so squaring halves the period from 2π to π: the negative arches of cos x are reflected up and repeat every π.