Precalculus Examples

Find the Inverse f(x) = square root of 16-x^2
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Rewrite the equation as .
Step 3.2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3.3
Simplify each side of the equation.
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Step 3.3.1
Use to rewrite as .
Step 3.3.2
Simplify the left side.
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Step 3.3.2.1
Simplify .
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Step 3.3.2.1.1
Multiply the exponents in .
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Step 3.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.3.2.1.1.2
Cancel the common factor of .
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Step 3.3.2.1.1.2.1
Cancel the common factor.
Step 3.3.2.1.1.2.2
Rewrite the expression.
Step 3.3.2.1.2
Simplify.
Step 3.4
Solve for .
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Step 3.4.1
Subtract from both sides of the equation.
Step 3.4.2
Divide each term in by and simplify.
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Step 3.4.2.1
Divide each term in by .
Step 3.4.2.2
Simplify the left side.
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Step 3.4.2.2.1
Dividing two negative values results in a positive value.
Step 3.4.2.2.2
Divide by .
Step 3.4.2.3
Simplify the right side.
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Step 3.4.2.3.1
Simplify each term.
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Step 3.4.2.3.1.1
Move the negative one from the denominator of .
Step 3.4.2.3.1.2
Rewrite as .
Step 3.4.2.3.1.3
Divide by .
Step 3.4.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4.4
Simplify .
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Step 3.4.4.1
Simplify the expression.
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Step 3.4.4.1.1
Rewrite as .
Step 3.4.4.1.2
Reorder and .
Step 3.4.4.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.4.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.4.5.1
First, use the positive value of the to find the first solution.
Step 3.4.5.2
Next, use the negative value of the to find the second solution.
Step 3.4.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
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Step 5.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 5.2
Find the range of .
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Step 5.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 5.3
Find the domain of .
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Step 5.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5.3.2
Solve for .
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Step 5.3.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.3.2.2
Set equal to and solve for .
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Step 5.3.2.2.1
Set equal to .
Step 5.3.2.2.2
Subtract from both sides of the equation.
Step 5.3.2.3
Set equal to and solve for .
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Step 5.3.2.3.1
Set equal to .
Step 5.3.2.3.2
Solve for .
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Step 5.3.2.3.2.1
Subtract from both sides of the equation.
Step 5.3.2.3.2.2
Divide each term in by and simplify.
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Step 5.3.2.3.2.2.1
Divide each term in by .
Step 5.3.2.3.2.2.2
Simplify the left side.
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Step 5.3.2.3.2.2.2.1
Dividing two negative values results in a positive value.
Step 5.3.2.3.2.2.2.2
Divide by .
Step 5.3.2.3.2.2.3
Simplify the right side.
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Step 5.3.2.3.2.2.3.1
Divide by .
Step 5.3.2.4
The final solution is all the values that make true.
Step 5.3.2.5
Use each root to create test intervals.
Step 5.3.2.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 5.3.2.6.1
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.2.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.6.1.2
Replace with in the original inequality.
Step 5.3.2.6.1.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 5.3.2.6.2
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.2.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.6.2.2
Replace with in the original inequality.
Step 5.3.2.6.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.2.6.3
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.2.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.6.3.2
Replace with in the original inequality.
Step 5.3.2.6.3.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 5.3.2.6.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 5.3.2.7
The solution consists of all of the true intervals.
Step 5.3.3
The domain is all values of that make the expression defined.
Step 5.4
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 6