Find the Inverse f(x)=|x-4|+1

$f (x) = | x - 4 | + 1$

Step 1

Write $f (x) = | x - 4 | + 1$ as an equation.

$y = | x - 4 | + 1$

Step 2

Interchange the variables.

$x = | y - 4 | + 1$

Step 3

Solve for $y$ .

Rewrite the equation as $| y - 4 | + 1 = x$ .

$| y - 4 | + 1 = x$

Subtract $1$ from both sides of the equation.

$| y - 4 | = x - 1$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 4 = \pm (x - 1)$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y - 4 = x - 1$

Move all terms not containing $y$ to the right side of the equation.

Add $4$ to both sides of the equation.

$y = x - 1 + 4$

Add $- 1$ and $4$ .

$y = x + 3$

Next, use the negative value of the $\pm$ to find the second solution.

$y - 4 = - (x - 1)$

Simplify $- (x - 1)$ .

Rewrite.

$y - 4 = 0 + 0 - (x - 1)$

Simplify by adding zeros.

$y - 4 = - (x - 1)$

Apply the distributive property.

$y - 4 = - x - - 1$

Multiply $- 1$ by $- 1$ .

$y - 4 = - x + 1$

Move all terms not containing $y$ to the right side of the equation.

Add $4$ to both sides of the equation.

$y = - x + 1 + 4$

Add $1$ and $4$ .

$y = - x + 5$

The complete solution is the result of both the positive and negative portions of the solution.

$y = x + 3$

$y = - x + 5$

$y = x + 3$

$y = - x + 5$

$y = x + 3$

$y = - x + 5$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = x + 3, - x + 5$

Step 5

Verify if $f^{- 1} (x) = x + 3, - x + 5$ is the inverse of $f (x) = | x - 4 | + 1$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = | x - 4 | + 1$ and $f^{- 1} (x) = x + 3, - x + 5$ and compare them.

Find the range of $f (x) = | x - 4 | + 1$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[1, \infty)$

Find the domain of $x + 3$ .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = x + 3, - x + 5$ is not equal to the range of $f (x) = | x - 4 | + 1$ , then $f^{- 1} (x) = x + 3, - x + 5$ is not an inverse of $f (x) = | x - 4 | + 1$ .

There is no inverse

Step 6

Find the Inverse f(x)=|x-4|+1

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