Question

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

\(f\left( x \right) = 4 + 3x - {x^2}\), \(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\)

Short answer

The value of the expression \(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\)is\( - 3 - h\).

Step-by-step solution

Describe the given information

The given function is as follows:

\(f\left( x \right) = 4 + 3x - {x^2}\)

It is required to find\(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\).

Simplify the expression \(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\)

Solve for\(f\left( {3 + h} \right)\).

\(\begin{aligned}f\left( {3 + h} \right) &= 4 + 3\left( {3 + h} \right) - {\left( {3 + h} \right)^2}\\ &= 4 + 9 + 3h - \left( {9 + 6h + {h^2}} \right)\\ &= 4 + 9 + 3h - 9 - 6h - {h^2}\\ &= 4 - 3h - {h^2}\end{aligned}\)

Solve for\(f\left( 3 \right)\).

\(\begin{aligned}f\left( 3 \right) &= 4 + 3\left( 3 \right) - {\left( 3 \right)^2}\\ &= 4 + 9 - 9\\ &= 4\end{aligned}\)

Substitute all the values in the expression\(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\).

\(\begin{aligned}\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h} &= \frac{{4 - 3h - {h^2} - 4}}{h}\\ &= \frac{{ - 3h - {h^2}}}{h}\\ &= - 3 - h\end{aligned}\)

Therefore, the value of the expression \(\frac{{f\left( {3 + h} \right) - f\left( 3 \right)}}{h}\)is\( - 3 - h\).