Evaluate the Function f(5/x)=(6(5/x)+2)/(4(5/x)+3)

$f (\frac{5}{x}) = \frac{6 (\frac{5}{x}) + 2}{4 (\frac{5}{x}) + 3}$

Step 1

Simplify the numerator.

Factor $2$ out of $6 \frac{5}{x} + 2$ .

Factor $2$ out of $6 \frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x})) + 2}{4 (\frac{5}{x}) + 3}$

Factor $2$ out of $2$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x})) + 2 (1)}{4 (\frac{5}{x}) + 3}$

Factor $2$ out of $2 (3 \frac{5}{x}) + 2 (1)$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x}) + 1)}{4 (\frac{5}{x}) + 3}$

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

Combine $3$ and $\frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (\frac{3 \cdot 5}{x} + 1)}{4 (\frac{5}{x}) + 3}$

Multiply $3$ by $5$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15}{x} + 1)}{4 (\frac{5}{x}) + 3}$

Write $1$ as a fraction with a common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15}{x} + \frac{x}{x})}{4 (\frac{5}{x}) + 3}$

Combine the numerators over the common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{4 (\frac{5}{x}) + 3}$

Step 2

Simplify the denominator.

Multiply $4 \frac{5}{x}$ .

Combine $4$ and $\frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{4 \cdot 5}{x} + 3}$

Multiply $4$ by $5$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20}{x} + 3}$

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

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20}{x} + \frac{3 x}{x}}$

Combine the numerators over the common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20 + 3 x}{x}}$

Step 3

Combine $2$ and $\frac{15 + x}{x}$ .

$f (\frac{5}{x}) = \frac{\frac{2 (15 + x)}{x}}{\frac{20 + 3 x}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5}{x}) = \frac{2 (15 + x)}{x} \cdot \frac{x}{20 + 3 x}$

Step 5

Cancel the common factor of $x$ .

Cancel the common factor.

$f (\frac{5}{x}) = \frac{2 (15 + x)}{x} \cdot \frac{x}{20 + 3 x}$

Rewrite the expression.

$f (\frac{5}{x}) = 2 (15 + x) (\frac{1}{20 + 3 x})$

Step 6

Combine $\frac{1}{20 + 3 x}$ and $2$ .

$f (\frac{5}{x}) = \frac{2}{20 + 3 x} \cdot (15 + x)$

Step 7

Multiply $\frac{2}{20 + 3 x}$ by $15 + x$ .

$f (\frac{5}{x}) = \frac{2 (15 + x)}{20 + 3 x}$

Step 8

Evaluate the Function f(5/x)=(6(5/x)+2)/(4(5/x)+3)

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