Question8. Given the function $f(x)=\left\{\begin{array}{lr} -2 x+4, & 0 \leq x<8 \\ -5 x+11, & x \geq 8 \end{array}\right.$ is the function increasing or decreasing over the interval $[2,7]$ ? Find the rate of change over this interval.

Studdy Solution

STEP 1

1. The function $f(x)$ is a piecewise function defined by different linear expressions over different intervals.
2. To determine if $f(x)$ is increasing or decreasing over the interval $[2,7]$ , we need to evaluate the behavior of the function within this interval.
3. The interval $[2,7]$ lies entirely within the first part of the piecewise function $-2x + 4$ .
4. The rate of change for a linear function $f(x) = mx + b$ is given by the slope $m$ .

STEP 2

1. Identify the function expression applicable over the interval $[2,7]$ .
2. Determine if this part of the function is increasing or decreasing.
3. Calculate the rate of change over the interval $[2,7]$ .

STEP 3

Identify the function expression applicable over the interval $[2,7]$ .
Given the function definition: $f(x)=\left\{\begin{array}{lr} -2 x+4, & 0 \leq x<8 \\ -5 x+11, & x \geq 8 \end{array}\right.$
The interval $[2,7]$ lies within $0 \leq x < 8$ , so we use the expression: $f(x) = -2x + 4$

STEP 4

Determine if the function $f(x) = -2x + 4$ is increasing or decreasing over the interval $[2,7]$ .
Since $f(x) = -2x + 4$ is a linear function with a slope of $-2$ , and the slope $-2$ is negative, the function is decreasing over any interval within its domain.

STEP 5

Calculate the rate of change over the interval $[2,7]$ .
For a linear function $f(x) = mx + b$ , the rate of change is given by the slope $m$ . Here, $m = -2$ .
Therefore, the rate of change over the interval $[2,7]$ is: $\text{Rate of change} = -2$
Solution: The function $f(x) = -2x + 4$ is decreasing over the interval $[2,7]$ . The rate of change over this interval is $-2$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms