### Home > CCA2 > Chapter 5 > Lesson 5.2.5 > Problem 5-121

Consider two functions *f*(*x*) = log *x* and *5-121 HW eTool* (Desmos). Homework Help ✎

Use these two functions to write an equation for a composite function and sketch its graph.

Use these two functions to write a different composite function and sketch its graph.

What makes the two composite functions so different from each other?

**Challenge:**Now try graphing*g*(*f*(*g*(*x*))).

Use the eTool below to view the graph of the equations and solve the parts of the problem.

Click the link at right for the full version of the eTool: *CCA2 5-121 HW eTool*

A composition of two functions is when the output of the first function becomes the input of the second.

*g*(*f*(*x*)) = |log(*x*)| or *f*(*g*(*x*)) = log|*x*|

See part (a).

Remember that the order of composition matters.

When using the absolute value first, you can use both positive and negative numbers for *x*, so the domain is different from the domain for the log function.When using the log first and then taking the absolute value, only positive values can be used for *x*. If the result of the log is negative, the absolute value makes it positive, so the range is only positive numbers.

Think this through; you can do it!