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Exponential function f is represented by the table.

x -2 -1 0 1 2
f(x) -46 -22 -10 -4 -1

Function g is represented by the equation.

Which statement correctly compares the two functions on the interval [-1, 2]?

A.
Both functions are increasing, but function g increases at a faster average rate.
B.
Both functions are increasing, but function f increases at a faster average rate.
C.
Only function f is increasing, but both functions are negative.
D.
Only function f is increasing, and only function f is negative.

1 answer

To compare the two functions \( f \) and \( g \) on the interval \([-1, 2]\), we need to analyze the behavior of function \( f \) based on the values provided in the table and consider the behavior of function \( g \) as described by its equation, which unfortunately is not provided in your question.

Let's first analyze the values of \( f(x) \):

For \( f(x) \):

  • At \( x = -1 \), \( f(-1) = -22 \)
  • At \( x = 0 \), \( f(0) = -10 \)
  • At \( x = 1 \), \( f(1) = -4 \)
  • At \( x = 2 \), \( f(2) = -1 \)

Now, let's determine if \( f \) is increasing on the interval \([-1, 2]\):

  • The value increases from \( -22 \) at \( x = -1 \) to \( -10 \) at \( x = 0 \) (increasing)
  • The value increases from \( -10 \) at \( x = 0 \) to \( -4 \) at \( x = 1 \) (increasing)
  • The value increases from \( -4 \) at \( x = 1 \) to \( -1 \) at \( x = 2 \) (increasing)

Thus, function \( f \) is indeed increasing on the interval \([-1, 2]\).

Next, we need to know the behavior of function \( g \). Since its equation was not provided in your question, we can only evaluate that function if given specific context or comparison regarding its properties.

If we assume function \( g \) is either constant, decreasing, or increasing based upon its equation, we could compare with \( f \).

Based on the increasing property of \( f \), if \( g \) is constant or decreasing, then option C ("Only function f is increasing, but both functions are negative") could potentially be correct.

Given that we cannot definitively determine the behavior of \( g \) without its equation, the best we can conclude from the information provided is:

  • Function \( f \) is increasing over the interval
  • Values of \( f \) remain negative over the interval

Ultimately, without additional context about \( g \), we might lean towards option C if \( g \) does not increase, but if more information regarding \( g \) indicates it is also increasing, then we would have to consider the average rates as mentioned in options A and B.

If you provide the equation of function \( g \), I would be able to analyze more accurately!