Finite Math Examples

$P (x) = 15 \sqrt{x - 1} + 3$

Step 1

Write $P (x) = 15 \sqrt{x - 1} + 3$ as an equation.

$y = 15 \sqrt{x - 1} + 3$

Step 2

Interchange the variables.

$x = 15 \sqrt{y - 1} + 3$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $15 \sqrt{y - 1} + 3 = x$ .

$15 \sqrt{y - 1} + 3 = x$

Step 3.2

Subtract $3$ from both sides of the equation.

$15 \sqrt{y - 1} = x - 3$

Step 3.3

To remove the radical on the left side of the equation, square both sides of the equation.

${(15 \sqrt{y - 1})}^{2} = {(x - 3)}^{2}$

Step 3.4

Simplify each side of the equation.

Tap for more steps...

Step 3.4.1

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

${(15 {(y - 1)}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify ${(15 {(y - 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.1

Apply the product rule to $15 {(y - 1)}^{\frac{1}{2}}$ .

$15^{2} {({(y - 1)}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Step 3.4.2.1.2

Raise $15$ to the power of $2$ .

$225 {({(y - 1)}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Step 3.4.2.1.3

Multiply the exponents in ${({(y - 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.3.1

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

$225 {(y - 1)}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Step 3.4.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.2.1.3.2.1

Cancel the common factor.

$225 {(y - 1)}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Step 3.4.2.1.3.2.2

Rewrite the expression.

$225 {(y - 1)}^{1} = {(x - 3)}^{2}$

Step 3.4.2.1.4

Simplify.

$225 (y - 1) = {(x - 3)}^{2}$

Step 3.4.2.1.5

Apply the distributive property.

$225 y + 225 \cdot - 1 = {(x - 3)}^{2}$

Step 3.4.2.1.6

Multiply $225$ by $- 1$ .

$225 y - 225 = {(x - 3)}^{2}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Simplify ${(x - 3)}^{2}$ .

Tap for more steps...

Step 3.4.3.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$225 y - 225 = (x - 3) (x - 3)$

Step 3.4.3.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.4.3.1.2.1

Apply the distributive property.

$225 y - 225 = x (x - 3) - 3 (x - 3)$

Step 3.4.3.1.2.2

Apply the distributive property.

$225 y - 225 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Step 3.4.3.1.2.3

Apply the distributive property.

$225 y - 225 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Step 3.4.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.4.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.4.3.1.3.1.1

Multiply $x$ by $x$ .

$225 y - 225 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Step 3.4.3.1.3.1.2

Move $- 3$ to the left of $x$ .

$225 y - 225 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Step 3.4.3.1.3.1.3

Multiply $- 3$ by $- 3$ .

$225 y - 225 = x^{2} - 3 x - 3 x + 9$

Step 3.4.3.1.3.2

Subtract $3 x$ from $- 3 x$ .

$225 y - 225 = x^{2} - 6 x + 9$

Step 3.5

Solve for $y$ .

Tap for more steps...

Step 3.5.1

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.5.1.1

Add $225$ to both sides of the equation.

$225 y = x^{2} - 6 x + 9 + 225$

Step 3.5.1.2

Add $9$ and $225$ .

$225 y = x^{2} - 6 x + 234$

Step 3.5.2

Divide each term in $225 y = x^{2} - 6 x + 234$ by $225$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $225 y = x^{2} - 6 x + 234$ by $225$ .

$\frac{225 y}{225} = \frac{x^{2}}{225} + \frac{- 6 x}{225} + \frac{234}{225}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Cancel the common factor of $225$ .

Tap for more steps...

Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{225 y}{225} = \frac{x^{2}}{225} + \frac{- 6 x}{225} + \frac{234}{225}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{225} + \frac{- 6 x}{225} + \frac{234}{225}$

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Simplify each term.

Tap for more steps...

Step 3.5.2.3.1.1

Cancel the common factor of $- 6$ and $225$ .

Tap for more steps...

Step 3.5.2.3.1.1.1

Factor $3$ out of $- 6 x$ .

$y = \frac{x^{2}}{225} + \frac{3 (- 2 x)}{225} + \frac{234}{225}$

Step 3.5.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.1.1.2.1

Factor $3$ out of $225$ .

$y = \frac{x^{2}}{225} + \frac{3 (- 2 x)}{3 (75)} + \frac{234}{225}$

Step 3.5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{225} + \frac{3 (- 2 x)}{3 \cdot 75} + \frac{234}{225}$

Step 3.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{225} + \frac{- 2 x}{75} + \frac{234}{225}$

Step 3.5.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{234}{225}$

Step 3.5.2.3.1.3

Cancel the common factor of $234$ and $225$ .

Tap for more steps...

Step 3.5.2.3.1.3.1

Factor $9$ out of $234$ .

$y = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{9 (26)}{225}$

Step 3.5.2.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.1.3.2.1

Factor $9$ out of $225$ .

$y = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{9 \cdot 26}{9 \cdot 25}$

Step 3.5.2.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{9 \cdot 26}{9 \cdot 25}$

Step 3.5.2.3.1.3.2.3

Rewrite the expression.

$y = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}$

Step 4

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

$P^{- 1} (x) = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}$

Step 5

Verify if $P^{- 1} (x) = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}$ is the inverse of $P (x) = 15 \sqrt{x - 1} + 3$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $P^{- 1} (15 \sqrt{x - 1} + 3)$ by substituting in the value of $P$ into $P^{- 1}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(15 \sqrt{x - 1} + 3)}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3

Simplify terms.

Tap for more steps...

Step 5.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.1.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1.1

Factor $3$ out of $15 \sqrt{x - 1} + 3$ .

Tap for more steps...

Step 5.2.3.1.1.1.1

Factor $3$ out of $15 \sqrt{x - 1}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(3 (5 \sqrt{x - 1}) + 3)}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.1.1.2

Factor $3$ out of $3$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(3 (5 \sqrt{x - 1}) + 3 (1))}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.1.1.3

Factor $3$ out of $3 (5 \sqrt{x - 1}) + 3 (1)$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(3 (5 \sqrt{x - 1} + 1))}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.1.2

Apply the product rule to $3 (5 \sqrt{x - 1} + 1)$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{3^{2} {(5 \sqrt{x - 1} + 1)}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.1.3

Raise $3$ to the power of $2$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{9 {(5 \sqrt{x - 1} + 1)}^{2}}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.2

Cancel the common factor of $9$ and $225$ .

Tap for more steps...

Step 5.2.3.1.2.1

Factor $9$ out of $9 {(5 \sqrt{x - 1} + 1)}^{2}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{9 ({(5 \sqrt{x - 1} + 1)}^{2})}{225} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.1.2.2.1

Factor $9$ out of $225$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{9 {(5 \sqrt{x - 1} + 1)}^{2}}{9 \cdot 25} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.2.2.2

Cancel the common factor.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{9 {(5 \sqrt{x - 1} + 1)}^{2}}{9 \cdot 25} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2}}{25} - \frac{2 (15 \sqrt{x - 1} + 3)}{75} + \frac{26}{25}$

Step 5.2.3.1.3

Cancel the common factor of $15 \sqrt{x - 1} + 3$ and $75$ .

Tap for more steps...

Step 5.2.3.1.3.1

Factor $3$ out of $2 (15 \sqrt{x - 1} + 3)$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2}}{25} - \frac{3 (2 (5 \sqrt{x - 1} + 1))}{75} + \frac{26}{25}$

Step 5.2.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.1.3.2.1

Factor $3$ out of $75$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2}}{25} - \frac{3 (2 (5 \sqrt{x - 1} + 1))}{3 (25)} + \frac{26}{25}$

Step 5.2.3.1.3.2.2

Cancel the common factor.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2}}{25} - \frac{3 (2 (5 \sqrt{x - 1} + 1))}{3 \cdot 25} + \frac{26}{25}$

Step 5.2.3.1.3.2.3

Rewrite the expression.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2}}{25} - \frac{2 (5 \sqrt{x - 1} + 1)}{25} + \frac{26}{25}$

Step 5.2.3.2

Combine the numerators over the common denominator.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2} - 2 (5 \sqrt{x - 1} + 1) + 26}{25}$

Step 5.2.3.3

Simplify each term.

Tap for more steps...

Step 5.2.3.3.1

Apply the distributive property.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2} - 2 (5 \sqrt{x - 1}) - 2 \cdot 1 + 26}{25}$

Step 5.2.3.3.2

Multiply $5$ by $- 2$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2} - 10 \sqrt{x - 1} - 2 \cdot 1 + 26}{25}$

Step 5.2.3.3.3

Multiply $- 2$ by $1$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2} - 10 \sqrt{x - 1} - 2 + 26}{25}$

Step 5.2.3.4

Add $- 2$ and $26$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{{(5 \sqrt{x - 1} + 1)}^{2} - 10 \sqrt{x - 1} + 24}{25}$

Step 5.2.4

Simplify the numerator.

Tap for more steps...

Step 5.2.4.1

Let $u = \sqrt{x - 1}$ . Substitute $u$ for all occurrences of $\sqrt{x - 1}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 u^{2} + 25}{25}$

Step 5.2.4.2

Factor $25$ out of $25 u^{2} + 25$ .

Tap for more steps...

Step 5.2.4.2.1

Factor $25$ out of $25 u^{2}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 (u^{2}) + 25}{25}$

Step 5.2.4.2.2

Factor $25$ out of $25$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 (u^{2}) + 25 (1)}{25}$

Step 5.2.4.2.3

Factor $25$ out of $25 (u^{2}) + 25 (1)$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 (u^{2} + 1)}{25}$

Step 5.2.4.3

Replace all occurrences of $u$ with $\sqrt{x - 1}$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ({\sqrt{x - 1}}^{2} + 1)}{25}$

Step 5.2.4.4

Simplify.

Tap for more steps...

Step 5.2.4.4.1

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

Tap for more steps...

Step 5.2.4.4.1.1

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

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ({({(x - 1)}^{\frac{1}{2}})}^{2} + 1)}{25}$

Step 5.2.4.4.1.2

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

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ({(x - 1)}^{\frac{1}{2} \cdot 2} + 1)}{25}$

Step 5.2.4.4.1.3

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

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ({(x - 1)}^{\frac{2}{2}} + 1)}{25}$

Step 5.2.4.4.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.4.4.1.4.1

Cancel the common factor.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ({(x - 1)}^{\frac{2}{2}} + 1)}{25}$

Step 5.2.4.4.1.4.2

Rewrite the expression.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 ((x - 1) + 1)}{25}$

Step 5.2.4.4.1.5

Simplify.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 (x - 1 + 1)}{25}$

Step 5.2.4.4.2

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

Tap for more steps...

Step 5.2.4.4.2.1

Add $- 1$ and $1$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 (x + 0)}{25}$

Step 5.2.4.4.2.2

Add $x$ and $0$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 x}{25}$

Step 5.2.5

Cancel the common factor of $25$ .

Tap for more steps...

Step 5.2.5.1

Cancel the common factor.

$P^{- 1} (15 \sqrt{x - 1} + 3) = \frac{25 x}{25}$

Step 5.2.5.2

Divide $x$ by $1$ .

$P^{- 1} (15 \sqrt{x - 1} + 3) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25})$ by substituting in the value of $P^{- 1}$ into $P$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{(\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) - 1} + 3$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25} - 1 \cdot \frac{25}{25}} + 3$

Step 5.3.3.2

Combine $- 1$ and $\frac{25}{25}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25} + \frac{- 1 \cdot 25}{25}} + 3$

Step 5.3.3.3

Combine the numerators over the common denominator.

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26 - 1 \cdot 25}{25}} + 3$

Step 5.3.3.4

Simplify the numerator.

Tap for more steps...

Step 5.3.3.4.1

Multiply $- 1$ by $25$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26 - 25}{25}} + 3$

Step 5.3.3.4.2

Subtract $25$ from $26$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{1}{25}} + 3$

Step 5.3.3.5

Factor using the perfect square rule.

Tap for more steps...

Step 5.3.3.5.1

Rewrite $225$ as $15^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{x^{2}}{15^{2}} - \frac{2 x}{75} + \frac{1}{25}} + 3$

Step 5.3.3.5.2

Rewrite $\frac{x^{2}}{15^{2}}$ as ${(\frac{x}{15})}^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15})}^{2} - \frac{2 x}{75} + \frac{1}{25}} + 3$

Step 5.3.3.5.3

Rewrite $1$ as $1^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15})}^{2} - \frac{2 x}{75} + \frac{1^{2}}{25}} + 3$

Step 5.3.3.5.4

Rewrite $25$ as $5^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15})}^{2} - \frac{2 x}{75} + \frac{1^{2}}{5^{2}}} + 3$

Step 5.3.3.5.5

Rewrite $\frac{1^{2}}{5^{2}}$ as ${(\frac{1}{5})}^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15})}^{2} - \frac{2 x}{75} + {(\frac{1}{5})}^{2}} + 3$

Step 5.3.3.5.6

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$\frac{2 x}{75} = 2 \cdot \frac{x}{15} \cdot \frac{1}{5}$

Step 5.3.3.5.7

Rewrite the polynomial.

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15})}^{2} - 2 \cdot \frac{x}{15} \cdot \frac{1}{5} + {(\frac{1}{5})}^{2}} + 3$

Step 5.3.3.5.8

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = \frac{x}{15}$ and $b = \frac{1}{5}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15} - \frac{1}{5})}^{2}} + 3$

Step 5.3.3.6

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15} - \frac{1}{5} \cdot \frac{3}{3})}^{2}} + 3$

Step 5.3.3.7

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

Tap for more steps...

Step 5.3.3.7.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15} - \frac{3}{5 \cdot 3})}^{2}} + 3$

Step 5.3.3.7.2

Multiply $5$ by $3$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x}{15} - \frac{3}{15})}^{2}} + 3$

Step 5.3.3.8

Combine the numerators over the common denominator.

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x - 3}{15})}^{2}} + 3$

Step 5.3.3.9

Apply the product rule to $\frac{x - 3}{15}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{{(x - 3)}^{2}}{15^{2}}} + 3$

Step 5.3.3.10

Raise $15$ to the power of $2$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{{(x - 3)}^{2}}{225}} + 3$

Step 5.3.3.11

Rewrite $225$ as $15^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{\frac{{(x - 3)}^{2}}{15^{2}}} + 3$

Step 5.3.3.12

Rewrite $\frac{{(x - 3)}^{2}}{15^{2}}$ as ${(\frac{x - 3}{15})}^{2}$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 \sqrt{{(\frac{x - 3}{15})}^{2}} + 3$

Step 5.3.3.13

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

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 (\frac{x - 3}{15}) + 3$

Step 5.3.3.14

Cancel the common factor of $15$ .

Tap for more steps...

Step 5.3.3.14.1

Cancel the common factor.

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = 15 (\frac{x - 3}{15}) + 3$

Step 5.3.3.14.2

Rewrite the expression.

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = x - 3 + 3$

Step 5.3.4

Combine the opposite terms in $x - 3 + 3$ .

Tap for more steps...

Step 5.3.4.1

Add $- 3$ and $3$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$P (\frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}) = x$

Step 5.4

Since $P^{- 1} (P (x)) = x$ and $P (P^{- 1} (x)) = x$ , then $P^{- 1} (x) = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}$ is the inverse of $P (x) = 15 \sqrt{x - 1} + 3$ .

$P^{- 1} (x) = \frac{x^{2}}{225} - \frac{2 x}{75} + \frac{26}{25}$