Question 6 of 10
What can you say about the -values of the two functions
and ?
A. has the largest possible -value.
B. and have equivalent maximum -values.
C. The maximum -value of approaches 2 .
D. has the largest possible -value.

Ask by May Ruiz. in the United States

Nov 07,2024

Question

Question 6 of 10
What can you say about the -values of the two functions
and ?
A. has the largest possible -value.
B. and have equivalent maximum -values.
C. The maximum -value of approaches 2 .
D. has the largest possible -value.

Ask by May Ruiz. in the United States

Nov 07,2024

UpStudy AI · Accepted Answer

To analyze the $$ y $$-values of the functions $$ f(x) = -5^x + 2 $$ and $$ g(x) = -5x^2 + 2 $$, let's examine each function:

1. **Function $$ f(x) = -5^x + 2 $$**:
   - As $$ x $$ increases, $$ 5^x $$ grows exponentially, making $$ -5^x $$ decrease rapidly. Therefore, $$ f(x) $$ approaches 2 as $$ x $$ approaches infinity.
   - As $$ x $$ approaches negative infinity, $$ 5^x $$ approaches 0, so $$ f(x) $$ approaches 2 from below.
   - The maximum $$ y $$-value of $$ f(x) $$ occurs at $$ x 	o -\infty $$ and is equal to 2.

2. **Function $$ g(x) = -5x^2 + 2 $$**:
   - This is a downward-opening parabola. The maximum $$ y $$-value occurs at the vertex.
   - The vertex of a parabola given by $$ ax^2 + bx + c $$ is at $$ x = -\frac{b}{2a} $$. Here, $$ a = -5 $$ and $$ b = 0 $$, so the vertex is at $$ x = 0 $$.
   - Evaluating $$ g(0) = -5(0)^2 + 2 = 2 $$. Thus, the maximum $$ y $$-value of $$ g(x) $$ is also 2.

Now, let's evaluate the options:

A. $$ g(x) $$ has the largest possible $$ y $$-value.  
- This is false; both functions have the same maximum $$ y $$-value of 2.

B. $$ f(x) $$ and $$ g(x) $$ have equivalent maximum $$ y $$-values.  
- This is true; both have a maximum $$ y $$-value of 2.

C. The maximum $$ y $$-value of $$ f(x) $$ approaches 2.  
- This is true, but it is not the most precise statement since the maximum $$ y $$-value is exactly 2.

D. $$ f(x) $$ has the largest possible $$ y $$-value.  
- This is false; both functions have the same maximum $$ y $$-value.

Based on this analysis, the best answer is:

**B. $$ f(x) $$ and $$ g(x) $$ have equivalent maximum $$ y $$-values.**
Both functions have the same maximum $$ y $$-value.

Question 6 of 10
What can you say about the -values of the two functions
and ?
A. has the largest possible -value.
B. and have equivalent maximum -values.
C. The maximum -value of approaches 2 .
D. has the largest possible -value.

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Question 6 of 10 What can you say about the -values of the two functions and ? A. has the largest possible -value. B. and have equivalent maximum -values. C. The maximum -value of approaches 2 . D. has the largest possible -value.

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Question 6 of 10
What can you say about the -values of the two functions
and ?
A. has the largest possible -value.
B. and have equivalent maximum -values.
C. The maximum -value of approaches 2 .
D. has the largest possible -value.