Composite Functions

Overview

Composition means applying one function after another.

The main idea is not substitution alone. The outputs of the first function must be acceptable inputs for the second function.

Use this branch with the hub Functions.

Meaning and Order

The composite

$$gf(x)=g(f(x))$$

means:

1. start with $x$;
2. apply $f$ first to get $f(x)$;
3. apply $g$ to that output.

So $gf$ is read from right to left in terms of operation order.

Order matters. In general,

$$gf\ne fg.$$

Example:

Let

$$f(x)=x+1,g(x)=x^2.$$

Then

$$gf(x)=g(f(x))=(x+1{)}^2,$$

but

$$fg(x)=f(g(x))=x^2+1.$$

These are not the same function.

Existence Condition

For $gf$ to exist on the whole domain of $f$, every output of $f$ must lie inside the domain of $g$:

$$R_f\subseteq D_g.$$

How to read this figure. Read the process from left to right but the notation from right to left: $x$ enters $f$ first and its output then enters $g$. For $gf$ to use every input in the original domain $D_f$, every value in $R_f$ must belong to $D_g$. If only some outputs are accepted, restrict the input set to those $x$ satisfying $f(x)\in D_g$.

If

$$R_f⊈D_g,$$

then $gf$ does not exist as a function on the full original domain of $f$.

If the question asks for a restricted composite, the repair is to restrict the domain of $f$ so that the surviving outputs of $f$ lie inside $D_g$.

Natural Domain of a Composite

Even when $R_f\subseteq D_g$ is not true, the expression $g(f(x))$ may still make sense for some inputs.

The natural domain is

$$D_{gf}=\{x\in D_f:f(x)\in D_g\}.$$

This formula says:

• start only with legal inputs for $f$;
• keep only those inputs whose $f$-outputs are legal inputs for $g$.

This distinguishes two conventions that must not be mixed:

• Composite on the whole stated domain of $f$: require $R_f\subseteq D_g$, giving $D_{gf}=D_f$.
• Composite expression on its natural domain: retain only $x\in D_f$ for which $f(x)\in D_g$.

State which interpretation you are using when the question wording leaves room for doubt.

Example:

Let

$$f(x)=x^2,g(x)=\ln x.$$

Then

$$D_f=\mathbb{R},D_g=(0,\infty ).$$

The composite is

$$gf(x)=\ln (x^2),$$

but it is defined only when

$$x^2>0.$$

So

$$D_{gf}=\mathbb{R}\setminus \{0\}.$$

Rule, Domain, and Range

To define a composite function properly:

1. check the order;
2. check the domain compatibility;
3. find the rule by substitution;
4. state the domain;
5. determine the range using the actual domain.

The range step is often missed.

Example:

Let

$$f(x)=2x+3,g(x)=\sqrt{x}.$$

Then

$$gf(x)=\sqrt{2x+3}.$$

Its domain is determined by

$$2x+3\ge 0,$$

so

$$D_{gf}=\left[-\frac{3}{2},\infty \right).$$

On this domain,

$$gf(x)\ge 0,$$

and every non-negative value is attained, so

$$R_{gf}=[0,\infty ).$$

Notice that the square-root restriction is imposed on the inside output $2x+3$, not directly on $x$ until the inequality is solved. This inside-to-outside reasoning works for roots, logarithms, denominators and other restricted outer functions.

Range of a Composite

There are two useful range viewpoints.

Method 1: Pass ranges through the functions

First find $R_f$. Then ask what $g$ does to the values in $R_f$.

This is especially useful when $R_f$ is a simple interval.

Example:

If

$$f(x)=-x^2,x\in \mathbb{R},$$

then

$$R_f=(-\infty ,0].$$

If

$$g(x)=e^x,$$

then applying $g$ to $(-\infty ,0]$ gives

$$(0,1].$$

So

$$R_{gf}=(0,1].$$

Method 2: Sketch or analyse the composite directly

Find

$$y=gf(x)$$

on its actual domain, then determine the range from the resulting function.

This is useful when the composite rule is simpler than reasoning through two separate graphs.

The same warning applies as before: do not determine range only from endpoints if the graph can turn inside the domain.

In set language, the range is the image of the actual composite domain:

$$R_{gf}=\{g(f(x)):x\in D_{gf}\}.$$

Although $R_{gf}\subseteq R_g$ always holds, equality need not hold because $g$ may receive only the subset $R_f\cap D_g$ of its possible inputs.

Restricting a Function So a Composite Exists

Sometimes a question asks for a restriction so that a composite exists.

The target condition is:

$$R_f\subseteq D_g.$$

If this fails and a composite is still required, reduce the domain of $f$ so that the resulting restricted range fits inside $D_g$.

Example:

Let

$$f(x)=x^2,g(x)=\ln x.$$

For $gf$ to exist on the chosen domain of $f$, we need

$$x^2>0.$$

The largest possible real domain for this composite expression is

$$\mathbb{R}\setminus \{0\}.$$

If the question requires an interval domain, we may choose either

$$(0,\infty )$$

or

$$(-\infty ,0).$$

The question wording decides the acceptable restriction.

When asked for a maximal domain, include every legal input, even when the answer is a union of intervals. An interval restriction such as $(0,\infty )$ is smaller than the maximal natural domain $\mathbb{R}\setminus \{0\}$ for $\ln (x^2)$.

Composition with an Inverse Function

If $f^{-1}$ exists, then

$$f^{-1}f(x)=x,x\in D_f,$$

and

$$f{f}^{-1}(x)=x,x\in R_f.$$

These expressions look the same algebraically, but they may represent identity functions on different domains.

This reinforces the main theme of the topic:

• a function is not just a rule;
• a function includes its domain.

For example, if $f:[-2,2]\to [-1,7]$ is defined by $f(x)=2x+3$, then

$$f^{-1}f:x\mapsto x,x\in [-2,2],$$

while

$$f{f}^{-1}:x\mapsto x,x\in [-1,7].$$

The rules agree, but the functions do not because their domains differ.

Exam Workflow

For a requested composite $gf$:

1. say explicitly that $f$ acts first;
2. record $D_f$, $R_f$ and $D_g$ if they are available;
3. check $R_f\subseteq D_g$ for a full-domain composite, or solve $f(x)\in D_g$ for the natural domain;
4. substitute to obtain $g(f(x))$;
5. simplify without losing restrictions;
6. find the range from the actual composite domain.

A correct simplified rule with an incorrect domain is an incorrect function.

Common Pitfalls

• Reading $gf$ as applying $g$ first. The function on the right acts first.
• Treating composition as substitution without checking domains.
• Assuming $R_f\subseteq D_g$ when only some outputs of $f$ are accepted by $g$.
• Forgetting to state the domain of the composite.
• Finding the range of a composite from the formula while ignoring the restricted domain.
• Assuming $gf=fg$.
• Assuming $f^{-1}f$ and $f{f}^{-1}$ are the same function without checking domains.

Related

• Functions
• Domains, Ranges, and One-One Functions
• Inverse Functions
• Graphing Techniques