Problem 88 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=7$$

Step-by-Step Solution

Answer

The simplified difference quotient for the given function is \(0\).

Step 1: Calculate \(f(x+h)\)

The expression \(f(x+h)\) is calculated by substituting \(x+h\) in place of \(x\) in the function s. The given function is \(f(x) = 7\). It is independent of \(x\), and so remains the same even after substitution. Hence, \(f(x+h) = 7\).

Step 2: Calculate \(f(x)\) and Substitute in the Difference Quotient

Now calculate \(f(x)\) for the function \(f(x) = 7\). As the function is a constant, this again gives us \(f(x) = 7\). Next, substitute the values of \(f(x+h)\) and \(f(x)\) into the difference quotient, giving us: \(\frac{f(x+h)-f(x)}{h} = \frac{7-7}{h}=0\)

Step 3: Simplify the Difference Quotient

Now we simplify the difference quotient. Since the numerator is zero, the quotient simplifies to \(0\).