QuestionDefine the absolute value function $f(x)=|4x+7|$ as a piecewise function.

Studdy Solution

STEP 1

Assumptions1. The function is an absolute value function. . The function can be broken down into piecewise-defined function with linear parts.

STEP 2

The absolute value function is defined as $|x| = \left\{\begin{array}{ll} -x, & x<0 \\ x, & x \geq0\end{array}\right.$

STEP 3

So, we can write the absolute value function $f(x)=|x+7|$ as a piecewise function by replacing $x$ with $(x+7)$ in the definition of the absolute value function.
$f(x) = \left\{\begin{array}{ll} -(x+7), &x+7<0 \\ x+7, &x+7 \geq0\end{array}\right.$

STEP 4

Now, we need to simplify the conditions in the piecewise function. For the first condition, we solve $4x+7<0$ for $x$ .

STEP 5

Subtract7 from both sides of the inequality $4x+7<0$ .
$4x < -7$

STEP 6

Divide both sides of the inequality $4x < -$ by4.
$x < -\frac{}{4}$

STEP 7

For the second condition, we solve $4x+7 \geq0$ for $x$ .

STEP 8

Subtract7 from both sides of the inequality $4x+7 \geq0$ .
$4x \geq -7$

STEP 9

Divide both sides of the inequality $4x \geq -7$ by4.
$x \geq -\frac{7}{4}$

STEP 10

Now we can write the absolute value function $f(x)=|4x+7|$ as a piecewise function with the simplified conditions.
$f(x) = \left\{\begin{array}{ll} -(4x+7), & x<-\frac{7}{4} \\ 4x+7, & x \geq -\frac{7}{4} \end{array}\right.$
So, the absolute value function $f(x)=|4x+7|$ can be written as a piecewise-defined function with linear parts as follows $f(x) = \left\{\begin{array}{ll} -(4x+7), & x<-\frac{7}{4} \\ 4x+7, & x \geq -\frac{7}{4} \end{array}\right.$

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