Question

For the function f(x)= x^2+9, construct and simplify the difference quotient f(x+h) -f(x) / h

Answer 1

We are given the following function:

`f\mleft(x\mright)=x^2+9`

We are asked to determine the following quotient:

`(f(x+h)-f(x))/(h)`

To do that, we will first determine the function f(x + h), to do that we will replace the variable "x" for the variable "x + h", like this:

`f(x+h)=(x+h)^2+9`

Solving the parenthesis, using the following property:

`(a+b)^2=a^2+2ab+b^2`

Applying the property:

`f(x+h)=x^2+2hx+h^2+9`

Substituting in the quotient:

`(x^2+2hx+h^2+9-(x^2+9))/(h)`

Now we change the sing to the terms inside the parenthesis since it is preceded by a minus sing:

`(x^2+2hx+h^2+9-x^2-9)/(h)`

Adding line terms:

`(2hx+h^2)/(h)`

Now we take "h" as a common factor in the numerator:

`(h(2x+h))/(h)`

Canceling out the "h":

`2x+h`