Algebra Examples

$f (x) = 25 x^{2} + 70 x + 49$

Step 1

Write $f (x) = 25 x^{2} + 70 x + 49$ as an equation.

$y = 25 x^{2} + 70 x + 49$

Step 2

Interchange the variables.

$x = 25 y^{2} + 70 y + 49$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $25 y^{2} + 70 y + 49 = x$ .

$25 y^{2} + 70 y + 49 = x$

Step 3.2

Subtract $x$ from both sides of the equation.

$25 y^{2} + 70 y + 49 - x = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = 25$ , $b = 70$ , and $c = 49 - x$ into the quadratic formula and solve for $y$ .

$\frac{- 70 \pm \sqrt{70^{2} - 4 \cdot (25 \cdot (49 - x))}}{2 \cdot 25}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $70$ to the power of $2$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4 \cdot 25 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.5.1.2

Multiply $- 4$ by $25$ .

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.5.1.3

Apply the distributive property.

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot 49 - 100 (- x)}}{2 \cdot 25}$

Step 3.5.1.4

Multiply $- 100$ by $49$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 - 100 (- x)}}{2 \cdot 25}$

Step 3.5.1.5

Multiply $- 1$ by $- 100$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 + 100 x}}{2 \cdot 25}$

Step 3.5.1.6

Subtract $4900$ from $4900$ .

$y = \frac{- 70 \pm \sqrt{0 + 100 x}}{2 \cdot 25}$

Step 3.5.1.7

Add $0$ and $100 x$ .

$y = \frac{- 70 \pm \sqrt{100 x}}{2 \cdot 25}$

Step 3.5.1.8

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 70 \pm \sqrt{10^{2} x}}{2 \cdot 25}$

Step 3.5.1.9

Pull terms out from under the radical.

$y = \frac{- 70 \pm 10 \sqrt{x}}{2 \cdot 25}$

Step 3.5.2

Multiply $2$ by $25$ .

$y = \frac{- 70 \pm 10 \sqrt{x}}{50}$

Step 3.5.3

Simplify $\frac{- 70 \pm 10 \sqrt{x}}{50}$ .

$y = \frac{- 7 \pm \sqrt{x}}{5}$

Step 3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $70$ to the power of $2$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4 \cdot 25 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.6.1.2

Multiply $- 4$ by $25$ .

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot 49 - 100 (- x)}}{2 \cdot 25}$

Step 3.6.1.4

Multiply $- 100$ by $49$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 - 100 (- x)}}{2 \cdot 25}$

Step 3.6.1.5

Multiply $- 1$ by $- 100$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 + 100 x}}{2 \cdot 25}$

Step 3.6.1.6

Subtract $4900$ from $4900$ .

$y = \frac{- 70 \pm \sqrt{0 + 100 x}}{2 \cdot 25}$

Step 3.6.1.7

Add $0$ and $100 x$ .

$y = \frac{- 70 \pm \sqrt{100 x}}{2 \cdot 25}$

Step 3.6.1.8

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 70 \pm \sqrt{10^{2} x}}{2 \cdot 25}$

Step 3.6.1.9

Pull terms out from under the radical.

$y = \frac{- 70 \pm 10 \sqrt{x}}{2 \cdot 25}$

Step 3.6.2

Multiply $2$ by $25$ .

$y = \frac{- 70 \pm 10 \sqrt{x}}{50}$

Step 3.6.3

Simplify $\frac{- 70 \pm 10 \sqrt{x}}{50}$ .

$y = \frac{- 7 \pm \sqrt{x}}{5}$

Step 3.6.4

Change the $\pm$ to $+$ .

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

Step 3.6.5

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 + \sqrt{x}}{5}$

Step 3.6.6

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

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

Step 3.6.7

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{x})$ .

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

Step 3.6.8

Move the negative in front of the fraction.

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

Step 3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $70$ to the power of $2$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4 \cdot 25 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.7.1.2

Multiply $- 4$ by $25$ .

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot (49 - x)}}{2 \cdot 25}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{- 70 \pm \sqrt{4900 - 100 \cdot 49 - 100 (- x)}}{2 \cdot 25}$

Step 3.7.1.4

Multiply $- 100$ by $49$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 - 100 (- x)}}{2 \cdot 25}$

Step 3.7.1.5

Multiply $- 1$ by $- 100$ .

$y = \frac{- 70 \pm \sqrt{4900 - 4900 + 100 x}}{2 \cdot 25}$

Step 3.7.1.6

Subtract $4900$ from $4900$ .

$y = \frac{- 70 \pm \sqrt{0 + 100 x}}{2 \cdot 25}$

Step 3.7.1.7

Add $0$ and $100 x$ .

$y = \frac{- 70 \pm \sqrt{100 x}}{2 \cdot 25}$

Step 3.7.1.8

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 70 \pm \sqrt{10^{2} x}}{2 \cdot 25}$

Step 3.7.1.9

Pull terms out from under the radical.

$y = \frac{- 70 \pm 10 \sqrt{x}}{2 \cdot 25}$

Step 3.7.2

Multiply $2$ by $25$ .

$y = \frac{- 70 \pm 10 \sqrt{x}}{50}$

Step 3.7.3

Simplify $\frac{- 70 \pm 10 \sqrt{x}}{50}$ .

$y = \frac{- 7 \pm \sqrt{x}}{5}$

Step 3.7.4

Change the $\pm$ to $-$ .

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

Step 3.7.5

Factor $- 1$ out of $- 7 - \sqrt{x}$ .

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Step 3.7.5.1

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 - \sqrt{x}}{5}$

Step 3.7.5.2

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

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

Step 3.7.5.3

Factor $- 1$ out of $- 1 (7) - (\sqrt{x})$ .

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

Step 3.7.5.4

Rewrite $- 1 (7 + \sqrt{x})$ as $- (7 + \sqrt{x})$ .

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

Step 3.7.6

Move the negative in front of the fraction.

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

Step 3.8

The final answer is the combination of both solutions.

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

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

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

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = - \frac{7 - \sqrt{x}}{5}, - \frac{7 + \sqrt{x}}{5}$ is the inverse of $f (x) = 25 x^{2} + 70 x + 49$ .

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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 $f (x) = 25 x^{2} + 70 x + 49$ and $f^{- 1} (x) = - \frac{7 - \sqrt{x}}{5}, - \frac{7 + \sqrt{x}}{5}$ and compare them.

Step 5.2

Find the range of $f (x) = 25 x^{2} + 70 x + 49$ .

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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 $- \frac{7 - \sqrt{x}}{5}$ .

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Step 5.3.1

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

$x \geq 0$

Step 5.3.2

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

$[0, \infty)$

Step 5.4

Find the domain of $f (x) = 25 x^{2} + 70 x + 49$ .

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Step 5.4.1

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.

$(- \infty, \infty)$

Step 5.5

Since the domain of $f^{- 1} (x) = - \frac{7 - \sqrt{x}}{5}, - \frac{7 + \sqrt{x}}{5}$ is the range of $f (x) = 25 x^{2} + 70 x + 49$ and the range of $f^{- 1} (x) = - \frac{7 - \sqrt{x}}{5}, - \frac{7 + \sqrt{x}}{5}$ is the domain of $f (x) = 25 x^{2} + 70 x + 49$ , then $f^{- 1} (x) = - \frac{7 - \sqrt{x}}{5}, - \frac{7 + \sqrt{x}}{5}$ is the inverse of $f (x) = 25 x^{2} + 70 x + 49$ .

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

Step 6