Algebra Examples

$f (x) = \frac{3 x^{2} + x - 4}{2 x^{2} - 7 x}$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Simplify.

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 2.1.1.1

Multiply by $1$ .

$f (x) = \frac{3 x^{2} + 1 x - 4}{2 x^{2} - 7 x}$

Step 2.1.1.2

Rewrite $1$ as $- 3$ plus $4$

$f (x) = \frac{3 x^{2} + (- 3 + 4) x - 4}{2 x^{2} - 7 x}$

Step 2.1.1.3

Apply the distributive property.

$f (x) = \frac{3 x^{2} - 3 x + 4 x - 4}{2 x^{2} - 7 x}$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$f (x) = \frac{(3 x^{2} - 3 x) + 4 x - 4}{2 x^{2} - 7 x}$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$f (x) = \frac{3 x (x - 1) + 4 (x - 1)}{2 x^{2} - 7 x}$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$f (x) = \frac{(x - 1) (3 x + 4)}{2 x^{2} - 7 x}$

Step 2.2

Factor $x$ out of $2 x^{2} - 7 x$ .

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Step 2.2.1

Factor $x$ out of $2 x^{2}$ .

$f (x) = \frac{(x - 1) (3 x + 4)}{x (2 x) - 7 x}$

Step 2.2.2

Factor $x$ out of $- 7 x$ .

$f (x) = \frac{(x - 1) (3 x + 4)}{x (2 x) + x \cdot - 7}$

Step 2.2.3

Factor $x$ out of $x (2 x) + x \cdot - 7$ .

$f (x) = \frac{(x - 1) (3 x + 4)}{x (2 x - 7)}$

Step 3

Find $f (- x)$ .

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Step 3.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \frac{((- x) - 1) (3 (- x) + 4)}{(- x) (2 (- x) - 7)}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $- 1$ by $3$ .

$f (- x) = \frac{(- x - 1) (- 3 x + 4)}{- x (2 (- x) - 7)}$

Step 3.2.2

Multiply $- 1$ by $2$ .

$f (- x) = \frac{(- x - 1) (- 3 x + 4)}{- x (- 2 x - 7)}$

Step 3.2.3

Move the negative in front of the fraction.

$f (- x) = - \frac{(- x - 1) (- 3 x + 4)}{x (- 2 x - 7)}$

Step 3.3

Factor $- 1$ out of $- x$ .

$f (- x) = - \frac{(- (x) - 1) (- 3 x + 4)}{x (- 2 x - 7)}$

Step 3.4

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

$f (- x) = - \frac{(- (x) - 1 \cdot 1) (- 3 x + 4)}{x (- 2 x - 7)}$

Step 3.5

Factor $- 1$ out of $- (x) - 1 (1)$ .

$f (- x) = - \frac{- (x + 1) (- 3 x + 4)}{x (- 2 x - 7)}$

Step 3.6

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$f (- x) = - \frac{- 1 (x + 1) (- 3 x + 4)}{x (- 2 x - 7)}$

Step 3.7

Factor $- 1$ out of $- 3 x$ .

$f (- x) = - \frac{- 1 (x + 1) (- (3 x) + 4)}{x (- 2 x - 7)}$

Step 3.8

Rewrite $4$ as $- 1 (- 4)$ .

$f (- x) = - \frac{- 1 (x + 1) (- (3 x) - 1 \cdot - 4)}{x (- 2 x - 7)}$

Step 3.9

Factor $- 1$ out of $- (3 x) - 1 (- 4)$ .

$f (- x) = - \frac{- 1 (x + 1) (- (3 x - 4))}{x (- 2 x - 7)}$

Step 3.10

Simplify the expression.

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Step 3.10.1

Rewrite $- (3 x - 4)$ as $- 1 (3 x - 4)$ .

$f (- x) = - \frac{- 1 (x + 1) (- 1 (3 x - 4))}{x (- 2 x - 7)}$

Step 3.10.2

Multiply $- 1$ by $- 1$ .

$f (- x) = - \frac{1 (x + 1) (3 x - 4)}{x (- 2 x - 7)}$

Step 3.10.3

Multiply $x + 1$ by $1$ .

$f (- x) = - \frac{(x + 1) (3 x - 4)}{x (- 2 x - 7)}$

Step 3.11

Factor $- 1$ out of $- 2 x$ .

$f (- x) = - \frac{(x + 1) (3 x - 4)}{x (- (2 x) - 7)}$

Step 3.12

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

$f (- x) = - \frac{(x + 1) (3 x - 4)}{x (- (2 x) - 1 \cdot 7)}$

Step 3.13

Factor $- 1$ out of $- (2 x) - 1 (7)$ .

$f (- x) = - \frac{(x + 1) (3 x - 4)}{x (- (2 x + 7))}$

Step 3.14

Simplify the expression.

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Step 3.14.1

Rewrite $- (2 x + 7)$ as $- 1 (2 x + 7)$ .

$f (- x) = - \frac{(x + 1) (3 x - 4)}{x (- 1 (2 x + 7))}$

Step 3.14.2

Move the negative in front of the fraction.

$f (- x) = \frac{(x + 1) (3 x - 4)}{x (2 x + 7)}$

Step 3.14.3

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 (\frac{(x + 1) (3 x - 4)}{x (2 x + 7)})$

Step 3.14.4

Multiply $\frac{(x + 1) (3 x - 4)}{x (2 x + 7)}$ by $1$ .

$f (- x) = \frac{(x + 1) (3 x - 4)}{x (2 x + 7)}$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $\frac{(x + 1) (3 x - 4)}{x (2 x + 7)}$ $\neq$ $\frac{(x - 1) (3 x + 4)}{x (2 x - 7)}$ , the function is not even.

The function is not even

Step 5

A function is odd if $f (- x) = - f (x)$ .

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Step 5.1

Multiply $- 1$ by $\frac{(x - 1) (3 x + 4)}{x (2 x - 7)}$ .

$- f (x) = - \frac{(x - 1) (3 x + 4)}{x (2 x - 7)}$

Step 5.2

Since $\frac{(x + 1) (3 x - 4)}{x (2 x + 7)}$ $\neq$ $- \frac{(x - 1) (3 x + 4)}{x (2 x - 7)}$ , the function is not odd.

The function is not odd

Step 6

The function is neither odd nor even

Step 7

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 8

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Step 9

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Step 10