Q 28. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$

Step-by-Step Solution

Answer

Domain of the composite function $f\circ g$of $f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$ is $[2, \infty )$.

Step 1. Given information.

We have given,

$f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$

Step 2. Concept used.

Domain is the set of all input values where the function is well defined and real.

Step 3. Explanation.

We have,

$f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$

Here we have g(x) is radical function and the function g(x) is well defined and real for all values of x grater than equal to 2.

That is, 

$[2, \infty )$

Similarly domain of the function f(x) is polynomial function and domain of the polynomial function is set of all real numbers.

That is,

$(-\infty , \infty )$

Therefore domain of the composite function of $f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$ is the intersection of the domain of f(x) and domain of g(x).

So domain of composite function of $f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$ is

$[2, \infty ).$

Step 4. Conclusion.

Hence, domain of the composite function $f\circ g$of $f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2} is [2, \infty )$.