Algebra Examples

- f (2 (x - 2)) + 1

$- f (2 (x - 2)) + 1$

Step 1

Simplify.

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Step 1.1

Subtract

1

$1$ from both sides of the equation.

- 2 y x + 4 y = - 1

$- 2 y x + 4 y = - 1$

Step 1.2

Factor

2 y

$2 y$ out of

- 2 y x + 4 y

$- 2 y x + 4 y$ .

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Step 1.2.1

Factor

2 y

$2 y$ out of

- 2 y x

$- 2 y x$ .

2 y (- 1 x) + 4 y = - 1

$2 y (- 1 x) + 4 y = - 1$

Step 1.2.2

Factor

2 y

$2 y$ out of

4 y

$4 y$ .

2 y (- 1 x) + 2 y (2) = - 1

$2 y (- 1 x) + 2 y (2) = - 1$

Step 1.2.3

Factor

2 y

$2 y$ out of

2 y (- 1 x) + 2 y (2)

$2 y (- 1 x) + 2 y (2)$ .

2 y (- 1 x + 2) = - 1

$2 y (- 1 x + 2) = - 1$

2 y (- 1 x + 2) = - 1

$2 y (- 1 x + 2) = - 1$

Step 1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

2 y (- x + 2) = - 1

$2 y (- x + 2) = - 1$

Step 1.4

Divide each term in

2 y (- x + 2) = - 1

$2 y (- x + 2) = - 1$ by

2 (- x + 2)

$2 (- x + 2)$ and simplify.

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Step 1.4.1

Divide each term in

2 y (- x + 2) = - 1

$2 y (- x + 2) = - 1$ by

2 (- x + 2)

$2 (- x + 2)$ .

\frac{2 y (- x + 2)}{2 (- x + 2)} = \frac{- 1}{2 (- x + 2)}

$\frac{2 y (- x + 2)}{2 (- x + 2)} = \frac{- 1}{2 (- x + 2)}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of

2

$2$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{2 y (- x + 2)}{2 (- x + 2)} = \frac{- 1}{2 (- x + 2)}$

Step 1.4.2.1.2

Rewrite the expression.

$\frac{y (- x + 2)}{- x + 2} = \frac{- 1}{2 (- x + 2)}$

Step 1.4.2.2

Cancel the common factor of

$- x + 2$ .

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Step 1.4.2.2.1

Cancel the common factor.

$\frac{y (- x + 2)}{- x + 2} = \frac{- 1}{2 (- x + 2)}$

Step 1.4.2.2.2

Divide

$y$ by

$1$ .

$y = \frac{- 1}{2 (- x + 2)}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{2 (- x + 2)}$

Step 1.4.3.2

Factor

$- 1$ out of

$- x$ .

$y = - \frac{1}{2 (- (x) + 2)}$

Step 1.4.3.3

Rewrite

$2$ as

$- 1 (- 2)$ .

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

Step 1.4.3.4

Factor

$- 1$ out of

$- (x) - 1 (- 2)$ .

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

Step 1.4.3.5

Simplify the expression.

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Step 1.4.3.5.1

Rewrite

$- (x - 2)$ as

$- 1 (x - 2)$ .

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

Step 1.4.3.5.2

Move the negative in front of the fraction.

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

Step 1.4.3.5.3

Multiply

$- 1$ by

$- 1$ .

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

Step 1.4.3.5.4

Multiply

$\frac{1}{2 (x - 2)}$ by

$1$ .

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

Step 2

Find where the expression

$\frac{1}{2 (x - 2)}$ is undefined.

$x = 2$

Step 3

Consider the rational function

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

$n$ is the degree of the numerator and

$m$ is the degree of the denominator.

1. If

$n < m$ , then the x-axis,

$y = 0$ , is the horizontal asymptote.

2. If

$n = m$ , then the horizontal asymptote is the line

$y = \frac{a}{b}$ .

3. If

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find

$n$ and

$m$ .

$n = 0$

$m = 1$

Step 5

Since

$n < m$ , the x-axis,

$y = 0$ , is the horizontal asymptote.

$y = 0$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes:

$x = 2$

Horizontal Asymptotes:

$y = 0$

No Oblique Asymptotes

Step 8