Given the function f (x) = 2 - 2 and the graph of g(x) shown, which function has a greater average rate of change over the interval 0≤x≤2? | your answer must state the Average Rate of Chage for each function, and indicate which is greater.
Given the function f (x) = 2 - 2 and the graph of g(x) shown, which function has a greater average rate of change over the interval 0≤x≤2? | your answer must state the Average Rate of Chage for each function, and indicate which is greater.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Mathematical Functions and Their Rates of Change**
**Problem Statement:**
Given the function \( f(x) = 2^x - 2 \) and the graph of \( g(x) \) shown, which function has a greater average rate of change over the interval \( 0 \leq x \leq 2 \)? Your answer must state the Average Rate of Change for each function, and indicate which is greater.
**Graph Description:**
The graph provided represents the function \( g(x) \). It is a curve that appears to show an exponential trend starting from the origin (0,0) and rising steeply. The graph specifically covers the interval from \( x = 0 \) to approximately \( x = 8 \), with corresponding \( y \)-values increasing.
### Steps to Solve:
1. **Calculate the Average Rate of Change for \( f(x) = 2^x - 2 \):**
The average rate of change over an interval \([a, b]\) for a function \( f(x) \) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For \( f(x) = 2^x - 2 \) over the interval \([0, 2]\):
\[
f(2) = 2^2 - 2 = 4 - 2 = 2
\]
\[
f(0) = 2^0 - 2 = 1 - 2 = -1
\]
So, the average rate of change for \( f(x) \) is:
\[
\frac{f(2) - f(0)}{2 - 0} = \frac{2 - (-1)}{2 - 0} = \frac{3}{2} = 1.5
\]
2. **Determine the Average Rate of Change for \( g(x) \):**
We must identify the \( y \)-values at \( x = 0 \) and \( x = 2 \) from the graph.
From the graph:
\[
g(0) = 0
\]
\[
g(2) \approx 4
\]
So, the average](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45e4ee6c-a715-4803-9fbe-92a41a442b45%2Faa9dc506-bd8d-4a49-9a7e-8ffa1a642e23%2Fquo73a9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Functions and Their Rates of Change**
**Problem Statement:**
Given the function \( f(x) = 2^x - 2 \) and the graph of \( g(x) \) shown, which function has a greater average rate of change over the interval \( 0 \leq x \leq 2 \)? Your answer must state the Average Rate of Change for each function, and indicate which is greater.
**Graph Description:**
The graph provided represents the function \( g(x) \). It is a curve that appears to show an exponential trend starting from the origin (0,0) and rising steeply. The graph specifically covers the interval from \( x = 0 \) to approximately \( x = 8 \), with corresponding \( y \)-values increasing.
### Steps to Solve:
1. **Calculate the Average Rate of Change for \( f(x) = 2^x - 2 \):**
The average rate of change over an interval \([a, b]\) for a function \( f(x) \) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For \( f(x) = 2^x - 2 \) over the interval \([0, 2]\):
\[
f(2) = 2^2 - 2 = 4 - 2 = 2
\]
\[
f(0) = 2^0 - 2 = 1 - 2 = -1
\]
So, the average rate of change for \( f(x) \) is:
\[
\frac{f(2) - f(0)}{2 - 0} = \frac{2 - (-1)}{2 - 0} = \frac{3}{2} = 1.5
\]
2. **Determine the Average Rate of Change for \( g(x) \):**
We must identify the \( y \)-values at \( x = 0 \) and \( x = 2 \) from the graph.
From the graph:
\[
g(0) = 0
\]
\[
g(2) \approx 4
\]
So, the average
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