Trigonometry Examples

$y = 5 x - 1$

Step 1

Interchange the variables.

$x = 5 y - 1$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $5 y - 1 = x$ .

$5 y - 1 = x$

Step 2.2

Add $1$ to both sides of the equation.

$5 y = x + 1$

Step 2.3

Divide each term in $5 y = x + 1$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 y = x + 1$ by $5$ .

$\frac{5 y}{5} = \frac{x}{5} + \frac{1}{5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{1}{5}$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{5} + \frac{1}{5}$

Step 3

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

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

Step 4

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

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (5 x - 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Combine the opposite terms in $5 x - 1 + 1$ .

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Step 4.2.4.1

Add $- 1$ and $1$ .

$f^{- 1} (5 x - 1) = \frac{5 x + 0}{5}$

Step 4.2.4.2

Add $5 x$ and $0$ .

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

Step 4.2.5

Cancel the common factor of $5$ .

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Divide $x$ by $1$ .

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

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{x}{5} + \frac{1}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{5} + \frac{1}{5}) = 5 (\frac{x}{5} + \frac{1}{5}) - 1$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{x}{5} + \frac{1}{5}) = 5 (\frac{x}{5}) + 5 (\frac{1}{5}) - 1$

Step 4.3.3.2

Cancel the common factor of $5$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{x}{5} + \frac{1}{5}) = 5 (\frac{x}{5}) + 5 (\frac{1}{5}) - 1$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{x}{5} + \frac{1}{5}) = x + 5 (\frac{1}{5}) - 1$

Step 4.3.3.3

Cancel the common factor of $5$ .

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Step 4.3.3.3.1

Cancel the common factor.

$f (\frac{x}{5} + \frac{1}{5}) = x + 5 (\frac{1}{5}) - 1$

Step 4.3.3.3.2

Rewrite the expression.

$f (\frac{x}{5} + \frac{1}{5}) = x + 1 - 1$

Step 4.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 4.3.4.1

Subtract $1$ from $1$ .

$f (\frac{x}{5} + \frac{1}{5}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{x}{5} + \frac{1}{5}) = x$

Step 4.4

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

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