Trigonometry Examples

$y = | x | + 6$

Step 1

Interchange the variables.

$x = | y | + 6$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $| y | + 6 = x$ .

$| y | + 6 = x$

Step 2.2

Subtract $6$ from both sides of the equation.

$| y | = x - 6$

Step 2.3

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

$y = \pm (x - 6)$

Step 2.4

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

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Step 2.4.1

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

$y = x - 6$

Step 2.4.2

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

$y = - (x - 6)$

Step 2.4.3

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

$y = x - 6$

$y = - (x - 6)$

$y = x - 6$

$y = - (x - 6)$

$y = x - 6$

$y = - (x - 6)$

Step 3

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

$f^{- 1} (x) = x - 6, - (x - 6)$

Step 4

Verify if $f^{- 1} (x) = x - 6, - (x - 6)$ is the inverse of $f (x) = | x | + 6$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = | x | + 6$ and $f^{- 1} (x) = x - 6, - (x - 6)$ and compare them.

Step 4.2

Find the range of $f (x) = | x | + 6$ .

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Step 4.2.1

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

Interval Notation:

$[6, \infty)$

Step 4.3

Find the domain of $x - 6$ .

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Step 4.3.1

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.

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = x - 6, - (x - 6)$ is not equal to the range of $f (x) = | x | + 6$ , then $f^{- 1} (x) = x - 6, - (x - 6)$ is not an inverse of $f (x) = | x | + 6$ .

There is no inverse

Step 5