Question

question: 8
the function f(x) is represented by f(x) = x the graph of the function g(x) is graphed below.
which of the following intervals are both functions increasing?
a. (-infty < x < 2)
b. (-infty < x < infty)
c. (0 < x < 4)
d. (2 < x < infty)

Explanation:

Step1: Analyze \( f(x) = x^2 \) (assuming typo, should be \( x^2 \))

The function \( f(x) = x^2 \) is a parabola opening upwards with vertex at \( (0,0) \). It is decreasing on \( (-\infty, 0) \) and increasing on \( (0, \infty) \)? Wait, no, vertex at \( x = 0 \), so increasing on \( (0, \infty) \)? Wait, no, standard parabola \( y = x^2 \) has vertex at \( (0,0) \), decreasing for \( x < 0 \), increasing for \( x > 0 \). But the graph here has vertex at \( x = 2 \) (from the graph, the minimum is at \( x = 2 \)). So \( f(x) \) is a parabola with vertex at \( (2, -4) \) (assuming grid). So \( f(x) \) is increasing on \( (2, \infty) \).

Step2: Analyze \( g(x) \) (assuming \( g(x) = x \), linear function)

The function \( g(x) = x \) has a slope of 1, so it is increasing for all \( x \in (-\infty, \infty) \).

Step3: Find interval where both increase

We need an interval where \( f(x) \) (parabola) is increasing and \( g(x) \) (line) is increasing. \( g(x) \) is always increasing. \( f(x) \) (parabola with vertex at \( x = 2 \)) is increasing on \( (2, \infty) \). So check options:

• Option A: \( -\infty < x < 2 \): \( f(x) \) is decreasing here (since vertex at 2, left of vertex is decreasing), \( g(x) \) is increasing. Not both.
• Option B: \( -\infty < x < \infty \): \( f(x) \) is decreasing on \( (-\infty, 2) \), so not both increasing.
• Option C: \( 0 < x < 4 \): \( f(x) \) is decreasing on \( 0 < x < 2 \), increasing on \( 2 < x < 4 \). So not all of this interval has \( f(x) \) increasing.
• Option D: \( 2 < x < \infty \): \( f(x) \) is increasing (right of vertex), \( g(x) \) is increasing (slope 1). So both increase here.

Answer:

D. \( 2 < x < \infty \)