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Find the difference quotient of \(\displaystyle f(x)=2 x - 6\) .

The difference quotient is \(\displaystyle \frac{f(x+h)-f(x)}{h}\). Evaluating \(\displaystyle f(x+h)=2 h + 2 x - 6\) and expanding gives \(\displaystyle f(x+h)=2 h + 2 x - 6\) Evaluating the difference quotient gives \(\displaystyle \frac{(2 h + 2 x - 6)-(2 x - 6)}{h}\) Simplifying gives \(\displaystyle \frac{2 h}{h}=2\)

Download \(\LaTeX\) 

\begin{question}Find the difference quotient of $f(x)=2 x - 6$ . 
    \soln{9cm}{The difference quotient is $\frac{f(x+h)-f(x)}{h}$. Evaluating $f(x+h)=2 h + 2 x - 6$ and expanding gives $f(x+h)=2 h + 2 x - 6$  Evaluating the difference quotient gives $\frac{(2 h + 2 x - 6)-(2 x - 6)}{h}$ Simplifying gives $\frac{2 h}{h}=2$ }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the difference quotient of  <img class="equation_image" title=" \displaystyle f(x)=2 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D2%20x%20-%206%20" alt="LaTeX:  \displaystyle f(x)=2 x - 6 " data-equation-content=" \displaystyle f(x)=2 x - 6 " />  . </p> </p>
HTML for Canvas
<p> <p>The difference quotient is  <img class="equation_image" title=" \displaystyle \frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \frac{f(x+h)-f(x)}{h} " /> . Evaluating  <img class="equation_image" title=" \displaystyle f(x+h)=2 h + 2 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D2%20h%20%2B%202%20x%20-%206%20" alt="LaTeX:  \displaystyle f(x+h)=2 h + 2 x - 6 " data-equation-content=" \displaystyle f(x+h)=2 h + 2 x - 6 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=2 h + 2 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D2%20h%20%2B%202%20x%20-%206%20" alt="LaTeX:  \displaystyle f(x+h)=2 h + 2 x - 6 " data-equation-content=" \displaystyle f(x+h)=2 h + 2 x - 6 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(2 h + 2 x - 6)-(2 x - 6)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%282%20h%20%2B%202%20x%20-%206%29-%282%20x%20-%206%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(2 h + 2 x - 6)-(2 x - 6)}{h} " data-equation-content=" \displaystyle \frac{(2 h + 2 x - 6)-(2 x - 6)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{2 h}{h}=2 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B2%20h%7D%7Bh%7D%3D2%20" alt="LaTeX:  \displaystyle \frac{2 h}{h}=2 " data-equation-content=" \displaystyle \frac{2 h}{h}=2 " />  </p> </p>