There are many ways of combining functions to creat more complicated functions. We can combine them using algebraic operations (adding, subtracting, multiplying, and dividing), and can substitute one function into another to create a composition.

We can combine functions by adding, subtracting, multiplying or dividing.

Let

Then

Then

= | "twice sin(x) plus the quadratic" |
= | |
||

= | "one plus the quotient of the quadratic and sin(x)" |
= | |

If we substitute one function into another, we can generate functions in which one function can be thought of as "inside" the other.

Let

Then

Then

= | "three plus the square of sin(x)" |
= | |
||

= | "the sine of the quadratic" | = | |

When we are using the function notation *f*(*x*) and
something other than the independent variable alone appears in the
parentheses, we are being asked to form a new function by
substitution.

think |
||

the inside function is () |
||

the inside function is (2x) |
||

the inside function is (-x) |
||

the inside function is (x+h) |

Do not confuse substitution with performing algebraic operations on the function itself. The letters and symbols may be the same, but the meaning is very different.

**Example**

Even though *f*(3*x*) and 3*f*(*x*) each contain
the letter *f*, the number 3 and the letter *x*, they
describe different operations:

think | |
||

substitute (3x) into f |
|||

multiply f by 3 |

think | |
||

the cosine of (x squared) |
|||

the square of (cosine of x) |

think | |
||

the reciprocal of (x+h) |
|||

f(x) plus the constant h |

There are times when it is customary to omit parentheses when writing certain functions.

One common operation which requires both the composition and the
combination of functions comes up in the calculation of slope, where
we have:

The expression

is called a difference quotient.

Let

Find and simplify the difference quotient.

In problems like this, be careful to form the composition
Find and simplify the difference quotient.

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

precal: 12.1 - composition and subsituting

page created: Tue Mar 20 13:06:24 2018

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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan