Question

Find f(x+h), f(x+h)-f(x), and(f(x+h)-f(x))/hfor the function f(x) = 3x.

Answer 1

We have that the original function is the following:

`f(x)\text{ = 3}\cdot x`

Now we must calculate the following:

1. f (x+h)

`\begin{gathered} f(x+h)\text{ = 3}\cdot(x+h) \\ f(x+h)\text{ = 3}\cdot x\text{ + 3}\cdot h \end{gathered}`

2. f(x+h)-f(x)

`\begin{gathered} f\text{ (x+h) - f(x) = 3}\cdot(x+h)\text{ - 3}\cdot x \\ f(x+h)\text{ - f(x) = 3}\cdot x\text{ + 3}\cdot h\text{ - 3}\cdot x \\ f(x+h)\text{ - f(x) = 3}\cdot h \end{gathered}`

3. (f(x+h)-f(x))/h

`\begin{gathered} \text{\lbrack}f(x+h)-f(x)\rbrack/h\text{ =}(3\cdot(x+h)-3\cdot x)/(h) \\ \text{\lbrack}f(x+h)-f(x)\rbrack/h=\text{ }\frac{3\cdot x\text{ +3}\cdot h\text{ -3}\cdot x}{h} \\ \text{\lbrack}f(x+h)-f(x)\rbrack/h=\text{ }(3\cdot h)/(h)\text{ } \\ \text{\lbrack}f(x+h)-f(x)\rbrack/h\text{ = 3} \end{gathered}`