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The problem is:

If $f(x) = -4x + 5$, find $\dfrac{f(a + h) – f(a)}{h}$

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closed as off-topic by glace, Kirill, 900 sit-ups a day, Davide Giraudo, Hakim Jun 25 at 20:57

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What have you tried? For example, what is $f(a+h)$ for the function $f(x)$ you gave us? –  Tom Jun 25 at 19:27
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2 Answers 2

If someone says $f(\color{red}{\mathrm{something}})$, they are telling you to take the formula of $f(x)$, and everywhere that appears $x$, you put $\color{red}{\mathrm{something}}$ instead. For example, if $f(x) = x^2 + 1$, $f(\color{red}{a+h})$ would be $(\color{red}{a+h})^2 + 1$. Can you do the same with the function you have there? (:

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$$\eqalign{\tt f(\color{blue}x)=-4\color{blue}x+5 \ \ &\rlap{\rlap{=}=}{\ >}\tt \ \ f(\color{blue}{a+h})=-4\color{blue}{(a+h)}+5=-4a-4h+5 \\ &\rlap{\rlap{=}=}{\ >} \ \ \tt f(\color{blue}a)=-4\color{blue}a+5. }$$ Therefore: $$\eqalign{\tt \dfrac{f(a+h)-f(a)}{h} &= \tt \dfrac{-4a-4h+5-(-4a+5)}{h}\\ &=\tt\require{cancel}\dfrac{\cancel{-4a}-4h+\cancel5+\cancel{4a}\cancel{-5}}{h}\\ \tt &=\tt \dfrac{-4h}{h} =-4.\tag{$\tt h\neq0$}\\ }$$

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