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# Precalculus Examples

Step 1

Interchange the variables.

Step 2

Step 2.1

Rewrite the equation as .

Step 2.2

Add to both sides of the equation.

Step 2.3

Divide each term in by and simplify.

Step 2.3.1

Divide each term in by .

Step 2.3.2

Simplify the left side.

Step 2.3.2.1

Cancel the common factor of .

Step 2.3.2.1.1

Cancel the common factor.

Step 2.3.2.1.2

Divide by .

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

Step 2.5

Simplify .

Step 2.5.1

Combine the numerators over the common denominator.

Step 2.5.2

Rewrite as .

Step 2.5.3

Multiply by .

Step 2.5.4

Combine and simplify the denominator.

Step 2.5.4.1

Multiply by .

Step 2.5.4.2

Raise to the power of .

Step 2.5.4.3

Raise to the power of .

Step 2.5.4.4

Use the power rule to combine exponents.

Step 2.5.4.5

Add and .

Step 2.5.4.6

Rewrite as .

Step 2.5.4.6.1

Use to rewrite as .

Step 2.5.4.6.2

Apply the power rule and multiply exponents, .

Step 2.5.4.6.3

Combine and .

Step 2.5.4.6.4

Cancel the common factor of .

Step 2.5.4.6.4.1

Cancel the common factor.

Step 2.5.4.6.4.2

Rewrite the expression.

Step 2.5.4.6.5

Evaluate the exponent.

Step 2.5.5

Combine using the product rule for radicals.

Step 2.5.6

Reorder factors in .

Step 2.6

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

Step 2.6.1

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

Step 2.6.2

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

Step 2.6.3

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

Step 3

Replace with to show the final answer.

Step 4

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 and and compare them.

Step 4.2

Find the range of .

Step 4.2.1

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

Interval Notation:

Step 4.3

Find the domain of .

Step 4.3.1

Set the radicand in greater than or equal to to find where the expression is defined.

Step 4.3.2

Solve for .

Step 4.3.2.1

Divide each term in by and simplify.

Step 4.3.2.1.1

Divide each term in by .

Step 4.3.2.1.2

Simplify the left side.

Step 4.3.2.1.2.1

Cancel the common factor of .

Step 4.3.2.1.2.1.1

Cancel the common factor.

Step 4.3.2.1.2.1.2

Divide by .

Step 4.3.2.1.3

Simplify the right side.

Step 4.3.2.1.3.1

Divide by .

Step 4.3.2.2

Subtract from both sides of the inequality.

Step 4.3.3

The domain is all values of that make the expression defined.

Step 4.4

Find the domain of .

Step 4.4.1

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.

Step 4.5

Since the domain of is the range of and the range of is the domain of , then is the inverse of .

Step 5