Question

Evaluate the difference quotient for the given function. Simplify your answer.

\(f\left( x \right) = \frac{{x + 3}}{{x + 1}}\), \(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\)

Short answer

The value of the expression \(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\)is\( - \frac{1}{{x + 1}}\).

Step-by-step solution

Describe the given information

The given function is as follows:

\(f\left( x \right) = \frac{{x + 3}}{{x + 1}}\)

It is required to find\(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\).

Step 2: Simplify the expression\(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\)

Solve for\(f\left( 1 \right)\).

\(\begin{aligned}{c}f\left( 1 \right) &= \frac{{1 + 3}}{{1 + 1}}\\ &= \frac{4}{2}\\ &= 2\end{aligned}\)

Substitute all the values in the expression\(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\).

\(\begin{aligned}{c}\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}} &= \frac{{\frac{{x + 3}}{{x + 1}} - 2}}{{x - 1}}\\ &= \frac{{\frac{{x + 3 - 2x - 2}}{{x + 1}}}}{{x - 1}}\\ &= \frac{{1 - x}}{{\left( {x + 1} \right)\left( {x - 1} \right)}}\\ &= - \frac{1}{{x + 1}}\end{aligned}\)

Therefore, the value of the expression \(\frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\)is\( - \frac{1}{{x + 1}}\).