Problem 126 - Precalculus Essentials

Question

If \(f(x)=3 x+7,\) find \(\frac{f(a+h)-f(a)}{h}\)

Step-by-Step Solution

Answer

The solution to the expression \(\frac{f(a+h)-f(a)}{h}\) for \(f(x) = 3x + 7\) is '3'

Step 1: Find f(a+h) and f(a)

Substitute \(x = a+h\) into the function to get \(f(a+h) = 3(a+h) + 7\). Then substitute \(x = a\) into the function to get \(f(a) = 3a + 7\).

Step 2: Substitute f(a+h) and f(a) into the expression

Substitute \(f(a+h)\) and \(f(a)\) into the expression \(\frac{f(a+h)-f(a)}{h}\) to get \(\frac{3(a+h)+7 - (3a+7)}{h}\).

Step 3: Simplify the expression

Simplify the expression \(\frac{3(a+h)+7 - (3a+7)}{h}\). The '7's will cancel out and we'll have \(\frac{3h}{h}\). We can then simplify it further by dividing \(3h\) by \(h\) to get final answer as '3'.