Question

the functions f(x) and g(x) are shown below. select the function that is best described by the statement. column 1: the domain is -∞ < x < ∞; the range is y ≥ -2; an x-intercept of the function is -1; the y-intercept of the function is 3; the function is increasing between x = -1 and x = 2. options: f(x), g(x), both f(x) and g(x), neither f(x) and g(x)

Explanation:

Step1: Analyze Domain

For \( f(x) \): The graph extends left and right without breaks, so domain is \( -\infty < x < \infty \).
For \( g(x) \): The graph also extends left and right, domain is \( -\infty < x < \infty \). So "Both \( f(x) \) and \( g(x) \)" for domain.

Step2: Analyze Range

For \( f(x) \): The lowest \( y \)-value is \( -2 \), so range \( y \geq -2 \).
For \( g(x) \): The graph goes downwards, \( y \)-values can be less than \( -2 \) (e.g., left side), so range not \( y \geq -2 \). So \( f(x) \) for range.

Step3: Analyze x - intercept at \( x = -1 \)

For \( f(x) \): The graph crosses \( x \)-axis at \( x = -1 \).
For \( g(x) \): The \( x \)-intercept is at \( x = 2 \) (from graph), not \( -1 \). So \( f(x) \) for \( x \)-intercept \( -1 \).

Step4: Analyze y - intercept \( 3 \)

For \( f(x) \): \( y \)-intercept is \( -1 \) (crosses \( y \)-axis at \( (0, -1) \)).
For \( g(x) \): \( y \)-intercept is \( -1 \) (crosses \( y \)-axis at \( (0, -1) \)). Neither has \( y \)-intercept \( 3 \).

Step5: Analyze Increasing between \( x = -1 \) and \( x = 2 \)

For \( f(x) \): From \( x = -1 \) to \( x = 2 \), the graph first decreases then increases? Wait, \( f(x) \) has a vertex at \( x = 1 \) (approx). Wait, re - check: \( f(x) \) left segment: from \( x = -1 \) to vertex (around \( x = 1 \))? Wait, original \( f(x) \) graph: left line from \( (-\infty, \dots) \) to \( (-1, 0) \), then down to vertex at \( (1, -2) \)? Wait, no, the graph of \( f(x) \): the left part is from \( (-\infty, \dots) \) to \( (-1, 0) \), then down to \( (1, -2) \), then up. Wait, between \( x = -1 \) and \( x = 2 \): from \( x = -1 \) to \( x = 1 \), \( f(x) \) is decreasing (from \( 0 \) to \( -2 \)), then from \( x = 1 \) to \( x = 2 \), increasing (from \( -2 \) to \( -1 \)). So not entirely increasing. For \( g(x) \): it's a line with positive slope, so between \( x = -1 \) and \( x = 2 \), it's increasing. Wait, maybe I made a mistake earlier. Wait, \( g(x) \) is a linear function with positive slope. So between \( x = -1 \) and \( x = 2 \), \( g(x) \) is increasing. \( f(x) \): from \( x = -1 \) to \( x = 2 \), let's check points: at \( x = -1 \), \( f(x) = 0 \); at \( x = 1 \), \( f(x) = -2 \); at \( x = 2 \), \( f(x) = -1 \). So from \( -1 \) to \( 1 \), it's decreasing, from \( 1 \) to \( 2 \), increasing. So not increasing throughout. \( g(x) \): since it's linear with positive slope, it's increasing everywhere, so between \( x = -1 \) and \( x = 2 \), it's increasing. Wait, maybe the question's "increasing" is strictly increasing? So \( g(x) \) is increasing here. But let's go back to the original graphs.

(Note: For each row, the answer depends on the specific property. Since the question is a table with multiple rows, but the user might want each row's answer. But the problem says "Select the function... for each statement".)

Answer:

• Domain \( -\infty < x < \infty \): Both \( f(x) \) and \( g(x) \)
• Range \( y \geq -2 \): \( f(x) \)
• \( x \)-intercept \( -1 \): \( f(x) \)
• \( y \)-intercept \( 3 \): Neither \( f(x) \) and \( g(x) \)
• Increasing between \( x = -1 \) and \( x = 2 \): \( g(x) \) (assuming \( g(x) \) is linear with positive slope, so increasing in that interval)