Require the Inner Output to Fit the Outer Domain
A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Let f(u)=√u and g(x)=x−2. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠2 (C) x≤2 (D) x≥2.
- Independent check inputs: Let f(u)=√u and g(x)=x−3. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠3 (C) x≤3 (D) x≥3.
Learn the idea
Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range. A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain. A square-root outer function requires its incoming quantity to be nonnegative.
Here is the method on the supplied teaching case. The output of g must be in the square-root domain, so x−2≥0. Thus x≥2, including the endpoint where the root is zero.
Checking the inner function’s domain alone is insufficient. Translate the outer restriction into an inequality on the original input, and keep an inclusive endpoint when a zero radicand is allowed.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Let f(u)=√u and g(x)=x−2. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠2 (C) x≤2 (D) x≥2. Solution: The output of g must be in the square-root domain, so x−2≥0. Thus x≥2, including the endpoint where the root is zero. The final answer is x≥2. Checking the inner function’s domain alone is insufficient. Translate the outer restriction into an inequality on the original input, and keep an inclusive endpoint when a zero radicand is allowed.
Check your reasoning
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Helpful tips
- A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain.
- Checking the inner function’s domain alone is insufficient.
See the answer and explanation
x≥3
The output of g must be in the square-root domain, so x−3≥0. Thus x≥3, including the endpoint where the root is zero.
Take it further
Changed-input task: Let f(u)=√u and g(x)=x−9. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠9 (C) x≤9 (D) x≥9. Show the mathematical reason, not only the selected answer. Expected result: x≥9. The output of g must be in the square-root domain, so x−9≥0. Thus x≥9, including the endpoint where the root is zero.
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