Trigonometry Examples

$f (x) = \frac{5}{x^{2}}$

Step 1

Write $f (x) = \frac{5}{x^{2}}$ as an equation.

$y = \frac{5}{x^{2}}$

Step 2

Interchange the variables.

$x = \frac{5}{y^{2}}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{5}{y^{2}} = x$ .

$\frac{5}{y^{2}} = x$

Step 3.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y^{2}$

Step 3.3

Multiply each term in $\frac{5}{y^{2}} = x$ by $y^{2}$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{5}{y^{2}} = x$ by $y^{2}$ .

$\frac{5}{y^{2}} y^{2} = x y^{2}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{5}{y^{2}} y^{2} = x y^{2}$

Step 3.3.2.1.2

Rewrite the expression.

$5 = x y^{2}$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $x y^{2} = 5$ .

$x y^{2} = 5$

Step 3.4.2

Divide each term in $x y^{2} = 5$ by $x$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $x y^{2} = 5$ by $x$ .

$\frac{x y^{2}}{x} = \frac{5}{x}$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{x y^{2}}{x} = \frac{5}{x}$

Step 3.4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{5}{x}$

Step 3.4.3

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

$y = \pm \sqrt{\frac{5}{x}}$

Step 3.4.4

Simplify $\pm \sqrt{\frac{5}{x}}$ .

Tap for more steps...

Step 3.4.4.1

Rewrite $\sqrt{\frac{5}{x}}$ as $\frac{\sqrt{5}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{5}}{\sqrt{x}}$

Step 3.4.4.2

Multiply $\frac{\sqrt{5}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{5}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 3.4.4.3

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.4.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 3.4.4.3.2

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 3.4.4.3.3

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 3.4.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 3.4.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 3.4.4.3.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Tap for more steps...

Step 3.4.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 3.4.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{5} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 3.4.4.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{5} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 3.4.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{5} \sqrt{x}}{x^{1}}$

Step 3.4.4.3.6.5

Simplify.

$y = \pm \frac{\sqrt{5} \sqrt{x}}{x}$

Step 3.4.4.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{5 x}}{x}$

Step 3.4.5

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

Tap for more steps...

Step 3.4.5.1

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

$y = \frac{\sqrt{5 x}}{x}$

Step 3.4.5.2

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

$y = - \frac{\sqrt{5 x}}{x}$

Step 3.4.5.3

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

$y = \frac{\sqrt{5 x}}{x}$

$y = - \frac{\sqrt{5 x}}{x}$

$y = \frac{\sqrt{5 x}}{x}$

$y = - \frac{\sqrt{5 x}}{x}$

$y = \frac{\sqrt{5 x}}{x}$

$y = - \frac{\sqrt{5 x}}{x}$

$y = \frac{\sqrt{5 x}}{x}$

$y = - \frac{\sqrt{5 x}}{x}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$ is the inverse of $f (x) = \frac{5}{x^{2}}$ .

Tap for more steps...

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 $f (x) = \frac{5}{x^{2}}$ and $f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{5}{x^{2}}$ .

Tap for more steps...

Step 5.2.1

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

Interval Notation:

$(0, \infty)$

Step 5.3

Find the domain of $\frac{\sqrt{5 x}}{x}$ .

Tap for more steps...

Step 5.3.1

Set the radicand in $\sqrt{5 x}$ greater than or equal to $0$ to find where the expression is defined.

$5 x \geq 0$

Step 5.3.2

Divide each term in $5 x \geq 0$ by $5$ and simplify.

Tap for more steps...

Step 5.3.2.1

Divide each term in $5 x \geq 0$ by $5$ .

$\frac{5 x}{5} \geq \frac{0}{5}$

Step 5.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{0}{5}$

Step 5.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{5}$

Step 5.3.2.3

Simplify the right side.

Tap for more steps...

Step 5.3.2.3.1

Divide $0$ by $5$ .

$x \geq 0$

Step 5.3.3

Set the denominator in $\frac{\sqrt{5 x}}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.3.4

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

$(0, \infty)$

Step 5.4

Find the domain of $f (x) = \frac{5}{x^{2}}$ .

Tap for more steps...

Step 5.4.1

Set the denominator in $\frac{5}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 5.4.2

Solve for $x$ .

Tap for more steps...

Step 5.4.2.1

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

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 5.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.4.3

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

$(- \infty, 0) \cup (0, \infty)$

Step 5.5

Since the domain of $f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$ is the range of $f (x) = \frac{5}{x^{2}}$ and the range of $f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$ is the domain of $f (x) = \frac{5}{x^{2}}$ , then $f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$ is the inverse of $f (x) = \frac{5}{x^{2}}$ .

$f^{- 1} (x) = \frac{\sqrt{5 x}}{x}, - \frac{\sqrt{5 x}}{x}$

Step 6