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The graph of function f is shown.

The graph of an exponential function passes through (minus 0.25, 10), (0, 6), (5, minus 2) also intercepts the x-axis at 1 unit.

Function g is represented by the table.

x -1 0 1 2 3
g(x) 15 3 0

Which statement correctly compares the two functions?

A.
They have different x- and y-intercepts but the same end behavior as x approaches ∞.
B.
They have the same x- and y-intercepts.
C.
They have the same y-intercept and the same end behavior as x approaches ∞.
D.
They have the same x-intercept but different end behavior as x approaches ∞.

1 answer

To analyze the two functions \( f \) and \( g \), we can start by determining their intercepts and end behavior.

  1. Function \( f \):

    • It passes through the points (−0.25, 10) and (0, 6), meaning:
      • The y-intercept \( f(0) = 6 \).
    • It also intercepts the x-axis at 1 unit, meaning:
      • The x-intercept is at \( x = 1 \).
    • Since \( f \) is an exponential function, as \( x \to \infty \), \( f(x) \) will approach either 0 (if the base of the exponential function is between 0 and 1) or infinity (if the base is greater than 1). However, since \( f(1) \) is 0, indicating it could potentially be a decreasing function.

  2. Function \( g \):

    • From the table, we have the values:
      • \( g(-1) = 15 \)
      • \( g(0) = 3 \)
      • \( g(1) = 0 \)
    • This indicates that the y-intercept \( g(0) = 3 \) and the x-intercept \( g(1) = 0 \).
    • The values provided suggest that \( g(x) \) is decreasing as \( x \) increases from -1 to 1.

Now, comparing \( f \) and \( g \):

  • For x-intercepts:

    • Both functions have the same x-intercept at \( x = 1 \).

  • For y-intercepts:

    • Function \( f \) has a y-intercept of \( 6 \) (at \( x = 0 \)).
    • Function \( g \) has a y-intercept of \( 3 \) (at \( x = 0 \)).
    • These are different.

  • For end behavior:

    • Function \( f \) (being an exponential function) will approach either 0 or infinity as \( x \to \infty \), depending on the base.
    • Function \( g \) appears to be decreasing and approaches 0 (given that \( g(1) = 0 \), and the value \( g(0) = 3 \) is larger than \( g(1) = 0 \) suggests that as \( x \to \infty \), \( g(x) \) may continue to decrease, possibly approaching 0).

This leads us to the options:

A. They have different x- and y-intercepts but the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts and unclear about end behavior similarity)

B. They have the same x- and y-intercepts. (False, different y-intercepts)

C. They have the same y-intercept and the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts)

D. They have the same x-intercept but different end behavior as \( x \) approaches \( \infty \). (True, same x-intercept but different behaviors as explained)

The correct answer is D.