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

1 answer

To analyze the functions \( f \) and \( g \) based on the provided information:

  1. Exponential Function \( f \):

    • It passes through the points \((-0.25, 10)\), \((0, 6)\), and \( (5, -2) \).
    • The fact that it intercepts the x-axis at 1 indicates \( f(1) = 0 \).
    • An exponential function typically has the end behavior where it approaches 0 as \( x \) approaches \(-\infty\) and increases without bound as \( x \) approaches \( \infty \).
    • \( f \) has a y-intercept at \( f(0) = 6 \).

  2. Function \( g \) (given as a table):

    • The values are:
      • \( g(-1) = 15 \)
      • \( g(0) = 3 \)
      • \( g(1) = 0 \)
    • The function has an x-intercept at \( g(1) = 0 \).
    • Since \( g(0) = 3 \), the y-intercept is \( 3 \).
    • The values of \( g(x) \) are decreasing, but we don't have information about \( g(x) \) for \( x > 1 \) to determine its end behavior.

  3. Interceptions:

    • Both functions have the same x-intercept at \( x = 1 \).
    • The y-intercepts are different: \( f \) has a y-intercept of \( 6 \) while \( g \) has a y-intercept of \( 3 \).

  4. End Behavior:

    • As \( x \) approaches \( ∞\):
      • The function \( f \) (being an exponential function) will tend to \( ∞\).
      • The behavior of \( g(x) \) as \( x \to ∞ \) cannot be determined from the table alone. We only have values for \( x = -1, 0, 1, 2, 3 \).

Based on this analysis, we can conclude:

  • Statement A is incorrect because they have different y-intercepts.
  • Statement B is correct in that they have the same x-intercept but we cannot conclude that their end behavior is different due to lack of information about \( g(x) \) as \( x \to ∞\).
  • Statement C is incorrect as discussed above.
  • Statement D is incorrect because they do not have the same y-intercept.

Thus, none of the statements perfectly describe the comparison, but the closest is B, assuming we assert that \( g \) behaves differently in end behavior (since we do not have sufficient information on that front). So, the answer is:

B. They have the same x-intercept but different end behavior as x approaches ∞.