Determine the domain of a composite function

For a real-valued composite $g(f(x))$ , the domain consists of inputs $x$ for which $f(x)$ is defined and whose output $f(x)$ lies in the domain of $g$ ; it is not generally found by simply intersecting the domains of $f$ and $g$ . Determining it involves applying restrictions from denominators, even roots, logarithms, or other formulas and expressing the resulting set in interval, inequality, or set notation. The focus is on standard algebraic and graphical functions, excluding abstract-domain and complex-valued cases.

Detailed Explanation: Determine the domain of a composite function

To determine the domain of a composite function $g(f(x))$ :

Find where the inner function $f(x)$ is defined.
Find the domain restriction for the outer function $g$ .
Make sure the output of $f(x)$ satisfies the restriction for $g$ .
Write the allowed $x$ -values in interval or inequality notation.

Example

Find the domain of

g(f(x))=\frac{1}{\sqrt{x-1}-2},

where

f(x)=\sqrt{x-1} \quad\text{and}\quad g(u)=\frac{1}{u-2}.

Step 1: Restrict the inner function

The expression inside an even root must be nonnegative:

x-1\ge 0.

Therefore,

x\ge 1.

So far, the domain of $f$ is $[1,\infty)$ .

Step 2: Apply the restriction from the outer function

The denominator of $g(u)=\frac{1}{u-2}$ cannot equal zero. Thus,

u-2\ne 0,

so the input to $g$ must satisfy

u\ne 2.

In the composite function, the input to $g$ is $f(x)=\sqrt{x-1}$ . Therefore,

\sqrt{x-1}\ne 2.

Solve this restriction:

x-1\ne 4

and hence

x\ne 5.

Step 3: Combine the restrictions

The two restrictions are

x\ge 1 \quad\text{and}\quad x\ne 5.

Therefore, the domain is

\boxed{[1,5)\cup(5,\infty)}.

Notice that we did not simply intersect the domains of $f$ and $g$ . We had to check whether the output of $f$ was allowed as an input for $g$ .

Learn by doing: Determine the domain of a composite function

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Function Composition to Domain - Integer over Root of Linear to Number Line

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