Questions: Given f(x)=3x-6, (a) Find f(x+h) and simplify. (b) Find (f(x+h)-f(x))/h and simplify.

Given f(x)=3x-6, (a) Find f(x+h) and simplify. (b) Find (f(x+h)-f(x))/h and simplify.
Transcript text: Given $f(x)=3 x-6$, (a) Find $f(x+h)$ and simplify. (b) Find $\frac{f(x+h)-f(x)}{h}$ and simplify.
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Solution

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Solution Steps

Solution Approach

(a) To find \( f(x+h) \), substitute \( x+h \) into the function \( f(x) = 3x - 6 \). Simplify the expression by distributing and combining like terms.

(b) To find the difference quotient \(\frac{f(x+h)-f(x)}{h}\), first calculate \( f(x+h) - f(x) \) using the expression from part (a). Then divide the result by \( h \) and simplify.

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

To find \( f(x+h) \), we substitute \( x+h \) into the function \( f(x) = 3x - 6 \):

\[ f(x+h) = 3(x+h) - 6 = 3x + 3h - 6 \]

Thus, the simplified expression for \( f(x+h) \) is:

\[ f(x+h) = 3h + 3x - 6 \]

Step 2: Calculate the Difference Quotient

Next, we compute the difference quotient \(\frac{f(x+h) - f(x)}{h}\):

\[ f(x+h) - f(x) = (3h + 3x - 6) - (3x - 6) = 3h \]

Now, we divide by \( h \):

\[ \frac{f(x+h) - f(x)}{h} = \frac{3h}{h} = 3 \]

Final Answer

The results are:

  • \( f(x+h) = 3h + 3x - 6 \)
  • The difference quotient is \( 3 \)

Thus, the final answers are:

\[ \boxed{f(x+h) = 3h + 3x - 6} \] \[ \boxed{\text{Difference Quotient} = 3} \]

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