Questions: The function (f) is defined by (f(x)=frac2 x+6x+5). Find (f(5 y)). [ f(5 y)= ]

$The function (f) is defined by (f(x)=frac2 x+6x+5). Find (f(5 y)). [ f(5 y)= ]$

Transcript text: The function $f$ is defined by $f(x)=\frac{2 x+6}{x+5}$. Find $f(5 y)$. \[ f(5 y)= \]

Solution

Solution Steps

To find $ f(5y) $ for the function $ f(x) = \frac{2x + 6}{x + 5} $, we need to substitute $ 5y $ in place of $ x $ in the function. This involves replacing every instance of $ x $ in the expression with $ 5y $ and simplifying the resulting expression.

Step 1: Substitute $ 5y $ into the Function

We start with the function defined as: \[ f(x) = \frac{2x + 6}{x + 5} \] To find $ f(5y) $, we substitute $ 5y $ for $ x $: \[ f(5y) = \frac{2(5y) + 6}{5y + 5} \]

Step 2: Simplify the Expression

Now we simplify the expression: \[ f(5y) = \frac{10y + 6}{5y + 5} \] Next, we can factor out common terms in the numerator and the denominator: \[ f(5y) = \frac{2(5y + 3)}{5(y + 1)} \]