Solved on Oct 27, 2023

Find the slope of the function $y = |x| + 1$ .

STEP 1

Assumptions1. The function given is $y=|x|+1$ . We are required to find the slope of this function3. We understand that the absolute value function, $|x|$ , is a piecewise function defined as $x$ for $x \geq0$ and $-x$ for $x <0$
4. We understand that the slope of a function at a point is given by the derivative of the function at that point

STEP 2

We first write the function $y=|x|+1$ as a piecewise function. $y =\begin{cases}x +1 & \text{for } x \geq0 \\ -x +1 & \text{for } x <0\end{cases}$

STEP 3

We find the derivative of the function for $x \geq0$ .
$y' = \frac{d}{dx}(x+1)$

STEP 4

Calculate the derivative for $x \geq0$ .
$y' =1$

STEP 5

We find the derivative of the function for $x <0$ .
$y' = \frac{d}{dx}(-x+1)$

STEP 6

Calculate the derivative for $x <0$ .
$y' = -1$

STEP 7

We combine the results from steps4 and6 to write the slope of the function as a piecewise function.
$y' =\begin{cases}1 & \text{for } x >0 \\ -1 & \text{for } x <0\end{cases}$ Note The derivative at $x=0$ is undefined because the function $y=|x|+1$ is not differentiable at $x=0$ . This is because the function has a sharp turn at this point, and the derivative from the left does not equal the derivative from the right.
So, the slope of the function $y=|x|+1$ is1 for $x >0$ , -1 for $x <0$ , and undefined for $x=0$ .

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Find the slope of the function $y = |x| + 1$ .

STEP 1

STEP 2

We first write the function $y=|x|+1$ as a piecewise function. $y =\begin{cases}x +1 & \text{for } x \geq0 \\ -x +1 & \text{for } x <0\end{cases}$

STEP 3

We find the derivative of the function for $x \geq0$ .
$y' = \frac{d}{dx}(x+1)$

STEP 4

Calculate the derivative for $x \geq0$ .
$y' =1$

STEP 5

We find the derivative of the function for $x <0$ .
$y' = \frac{d}{dx}(-x+1)$

STEP 6

Calculate the derivative for $x <0$ .
$y' = -1$

STEP 7

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Find the slope of the function y=∣x∣+1y = |x| + 1y=∣x∣+1.

STEP 1

STEP 2

We first write the function y=∣x∣+1y=|x|+1y=∣x∣+1 as a piecewise function.y={x+1for x≥0−x+1for x<0y =\begin{cases}x +1 & \text{for } x \geq0 \\ -x +1 & \text{for } x <0\end{cases}y={x+1−x+1​for x≥0for x<0​

STEP 3

We find the derivative of the function for x≥0x \geq0x≥0.y′=ddx(x+1)y' = \frac{d}{dx}(x+1)y′=dxd​(x+1)

STEP 4

Calculate the derivative for x≥0x \geq0x≥0.y′=1y' =1y′=1

STEP 5

We find the derivative of the function for x<0x <0x<0.y′=ddx(−x+1)y' = \frac{d}{dx}(-x+1)y′=dxd​(−x+1)

STEP 6

Calculate the derivative for x<0x <0x<0.y′=−1y' = -1y′=−1

STEP 7

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Find the slope of the function $y = |x| + 1$ .

We first write the function $y=|x|+1$ as a piecewise function. $y =\begin{cases}x +1 & \text{for } x \geq0 \\ -x +1 & \text{for } x <0\end{cases}$

We find the derivative of the function for $x \geq0$ .
$y' = \frac{d}{dx}(x+1)$

Calculate the derivative for $x \geq0$ .
$y' =1$

We find the derivative of the function for $x <0$ .
$y' = \frac{d}{dx}(-x+1)$

Calculate the derivative for $x <0$ .
$y' = -1$