Trigonometry Examples

$f (x) = - \frac{7}{5} x + \frac{5}{3}$

Step 1

Write $f (x) = - \frac{7}{5} x + \frac{5}{3}$ as an equation.

$y = - \frac{7}{5} x + \frac{5}{3}$

Step 2

Interchange the variables.

$x = - \frac{7}{5} y + \frac{5}{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{7}{5} y + \frac{5}{3} = x$ .

$- \frac{7}{5} y + \frac{5}{3} = x$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $y$ and $\frac{7}{5}$ .

$- \frac{y \cdot 7}{5} + \frac{5}{3} = x$

Step 3.2.2

Move $7$ to the left of $y$ .

$- \frac{7 y}{5} + \frac{5}{3} = x$

Step 3.3

Subtract $\frac{5}{3}$ from both sides of the equation.

$- \frac{7 y}{5} = x - \frac{5}{3}$

Step 3.4

Multiply both sides of the equation by $- \frac{5}{7}$ .

$- \frac{5}{7} (- \frac{7 y}{5}) = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $- \frac{5}{7} (- \frac{7 y}{5})$ .

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Step 3.5.1.1.1

Cancel the common factor of $5$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{5}{7}$ into the numerator.

$\frac{- 5}{7} (- \frac{7 y}{5}) = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{7 y}{5}$ into the numerator.

$\frac{- 5}{7} \cdot \frac{- 7 y}{5} = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{7} \cdot \frac{- 7 y}{5} = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{7} \cdot \frac{- 7 y}{5} = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{7} (- 7 y) = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.2

Cancel the common factor of $7$ .

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Step 3.5.1.1.2.1

Factor $7$ out of $- 7 y$ .

$\frac{- 1}{7} (7 (- y)) = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{7} (7 (- y)) = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{5}{7} (x - \frac{5}{3})$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- \frac{5}{7} (x - \frac{5}{3})$ .

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Step 3.5.2.1.1

Apply the distributive property.

$y = - \frac{5}{7} x - \frac{5}{7} (- \frac{5}{3})$

Step 3.5.2.1.2

Combine $x$ and $\frac{5}{7}$ .

$y = - \frac{x \cdot 5}{7} - \frac{5}{7} (- \frac{5}{3})$

Step 3.5.2.1.3

Multiply $- \frac{5}{7} (- \frac{5}{3})$ .

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Step 3.5.2.1.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{x \cdot 5}{7} + 1 (\frac{5}{7}) \frac{5}{3}$

Step 3.5.2.1.3.2

Multiply $\frac{5}{7}$ by $1$ .

$y = - \frac{x \cdot 5}{7} + \frac{5}{7} \cdot \frac{5}{3}$

Step 3.5.2.1.3.3

Multiply $\frac{5}{7}$ by $\frac{5}{3}$ .

$y = - \frac{x \cdot 5}{7} + \frac{5 \cdot 5}{7 \cdot 3}$

Step 3.5.2.1.3.4

Multiply $5$ by $5$ .

$y = - \frac{x \cdot 5}{7} + \frac{25}{7 \cdot 3}$

Step 3.5.2.1.3.5

Multiply $7$ by $3$ .

$y = - \frac{x \cdot 5}{7} + \frac{25}{21}$

Step 3.5.2.1.4

Move $5$ to the left of $x$ .

$y = - \frac{5 x}{7} + \frac{25}{21}$

Step 4

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

$f^{- 1} (x) = - \frac{5 x}{7} + \frac{25}{21}$

Step 5

Verify if $f^{- 1} (x) = - \frac{5 x}{7} + \frac{25}{21}$ is the inverse of $f (x) = - \frac{7}{5} x + \frac{5}{3}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (- \frac{7}{5} x + \frac{5}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7}{5} x + \frac{5}{3})}{7} + \frac{25}{21}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Combine $x$ and $\frac{7}{5}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{x \cdot 7}{5} + \frac{5}{3})}{7} + \frac{25}{21}$

Step 5.2.3.1.2

Move $7$ to the left of $x$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 \cdot x}{5} + \frac{5}{3})}{7} + \frac{25}{21}$

Step 5.2.3.1.3

To write $- \frac{7 x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x}{5} \cdot \frac{3}{3} + \frac{5}{3})}{7} + \frac{25}{21}$

Step 5.2.3.1.4

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x}{5} \cdot \frac{3}{3} + \frac{5}{3} \cdot \frac{5}{5})}{7} + \frac{25}{21}$

Step 5.2.3.1.5

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1.5.1

Multiply $\frac{7 x}{5}$ by $\frac{3}{3}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x \cdot 3}{5 \cdot 3} + \frac{5}{3} \cdot \frac{5}{5})}{7} + \frac{25}{21}$

Step 5.2.3.1.5.2

Multiply $5$ by $3$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x \cdot 3}{15} + \frac{5}{3} \cdot \frac{5}{5})}{7} + \frac{25}{21}$

Step 5.2.3.1.5.3

Multiply $\frac{5}{3}$ by $\frac{5}{5}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x \cdot 3}{15} + \frac{5 \cdot 5}{3 \cdot 5})}{7} + \frac{25}{21}$

Step 5.2.3.1.5.4

Multiply $3$ by $5$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (- \frac{7 x \cdot 3}{15} + \frac{5 \cdot 5}{15})}{7} + \frac{25}{21}$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (\frac{- 7 x \cdot 3 + 5 \cdot 5}{15})}{7} + \frac{25}{21}$

Step 5.2.3.1.7

Simplify the numerator.

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Step 5.2.3.1.7.1

Multiply $3$ by $- 7$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (\frac{- 21 x + 5 \cdot 5}{15})}{7} + \frac{25}{21}$

Step 5.2.3.1.7.2

Multiply $5$ by $5$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{5 (\frac{- 21 x + 25}{15})}{7} + \frac{25}{21}$

Step 5.2.3.2

Combine $5$ and $\frac{- 21 x + 25}{15}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{\frac{5 (- 21 x + 25)}{15}}{7} + \frac{25}{21}$

Step 5.2.3.3

Reduce the expression $\frac{5 (- 21 x + 25)}{15}$ by cancelling the common factors.

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Step 5.2.3.3.1

Factor $5$ out of $15$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{\frac{5 (- 21 x + 25)}{5 \cdot 3}}{7} + \frac{25}{21}$

Step 5.2.3.3.2

Cancel the common factor.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{\frac{5 (- 21 x + 25)}{5 \cdot 3}}{7} + \frac{25}{21}$

Step 5.2.3.3.3

Rewrite the expression.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{\frac{- 21 x + 25}{3}}{7} + \frac{25}{21}$

Step 5.2.3.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - (\frac{- 21 x + 25}{3} \cdot \frac{1}{7}) + \frac{25}{21}$

Step 5.2.3.5

Multiply $\frac{- 21 x + 25}{3} \cdot \frac{1}{7}$ .

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Step 5.2.3.5.1

Multiply $\frac{- 21 x + 25}{3}$ by $\frac{1}{7}$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{- 21 x + 25}{3 \cdot 7} + \frac{25}{21}$

Step 5.2.3.5.2

Multiply $3$ by $7$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = - \frac{- 21 x + 25}{21} + \frac{25}{21}$

Step 5.2.4

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{- (- 21 x + 25) + 25}{21}$

Step 5.2.5

Simplify each term.

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Step 5.2.5.1

Apply the distributive property.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{- (- 21 x) - 1 \cdot 25 + 25}{21}$

Step 5.2.5.2

Multiply $- 21$ by $- 1$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{21 x - 1 \cdot 25 + 25}{21}$

Step 5.2.5.3

Multiply $- 1$ by $25$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{21 x - 25 + 25}{21}$

Step 5.2.6

Simplify terms.

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Step 5.2.6.1

Combine the opposite terms in $21 x - 25 + 25$ .

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Step 5.2.6.1.1

Add $- 25$ and $25$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{21 x + 0}{21}$

Step 5.2.6.1.2

Add $21 x$ and $0$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{21 x}{21}$

Step 5.2.6.2

Cancel the common factor of $21$ .

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Step 5.2.6.2.1

Cancel the common factor.

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = \frac{21 x}{21}$

Step 5.2.6.2.2

Divide $x$ by $1$ .

$f^{- 1} (- \frac{7}{5} x + \frac{5}{3}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (- \frac{5 x}{7} + \frac{25}{21})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = - \frac{7}{5} \cdot (- \frac{5 x}{7} + \frac{25}{21}) + \frac{5}{3}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{5 x}{7} + \frac{25}{21}) = - \frac{7}{5} \cdot (- \frac{5 x}{7}) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{7}{5}$ into the numerator.

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{- 7}{5} \cdot (- \frac{5 x}{7}) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.2.2

Move the leading negative in $- \frac{5 x}{7}$ into the numerator.

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{- 7}{5} \cdot \frac{- 5 x}{7} - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.2.3

Factor $7$ out of $- 7$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{7 (- 1)}{5} \cdot \frac{- 5 x}{7} - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.2.4

Cancel the common factor.

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{7 \cdot - 1}{5} \cdot \frac{- 5 x}{7} - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.2.5

Rewrite the expression.

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{- 1}{5} \cdot (- 5 x) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.3

Cancel the common factor of $5$ .

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Step 5.3.3.3.1

Factor $5$ out of $- 5 x$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{- 1}{5} \cdot (5 (- x)) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.3.2

Cancel the common factor.

$f (- \frac{5 x}{7} + \frac{25}{21}) = \frac{- 1}{5} \cdot (5 (- x)) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.3.3

Rewrite the expression.

$f (- \frac{5 x}{7} + \frac{25}{21}) = - 1 (- x) - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = 1 x - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x - \frac{7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.6

Cancel the common factor of $7$ .

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Step 5.3.3.6.1

Move the leading negative in $- \frac{7}{5}$ into the numerator.

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{- 7}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.6.2

Factor $7$ out of $- 7$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{7 (- 1)}{5} \cdot \frac{25}{21} + \frac{5}{3}$

Step 5.3.3.6.3

Factor $7$ out of $21$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{7 \cdot - 1}{5} \cdot \frac{25}{7 \cdot 3} + \frac{5}{3}$

Step 5.3.3.6.4

Cancel the common factor.

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{7 \cdot - 1}{5} \cdot \frac{25}{7 \cdot 3} + \frac{5}{3}$

Step 5.3.3.6.5

Rewrite the expression.

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{- 1}{5} \cdot \frac{25}{3} + \frac{5}{3}$

Step 5.3.3.7

Cancel the common factor of $5$ .

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Step 5.3.3.7.1

Factor $5$ out of $25$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{- 1}{5} \cdot \frac{5 (5)}{3} + \frac{5}{3}$

Step 5.3.3.7.2

Cancel the common factor.

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + \frac{- 1}{5} \cdot \frac{5 \cdot 5}{3} + \frac{5}{3}$

Step 5.3.3.7.3

Rewrite the expression.

$f (- \frac{5 x}{7} + \frac{25}{21}) = x - 1 (\frac{5}{3}) + \frac{5}{3}$

Step 5.3.3.8

Rewrite $- 1 (\frac{5}{3})$ as $- (\frac{5}{3})$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x - \frac{5}{3} + \frac{5}{3}$

Step 5.3.4

Combine the opposite terms in $x - \frac{5}{3} + \frac{5}{3}$ .

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Step 5.3.4.1

Add $- \frac{5}{3}$ and $\frac{5}{3}$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- \frac{5 x}{7} + \frac{25}{21}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5 x}{7} + \frac{25}{21}$ is the inverse of $f (x) = - \frac{7}{5} x + \frac{5}{3}$ .

$f^{- 1} (x) = - \frac{5 x}{7} + \frac{25}{21}$