If f(x)=√(x^2+16), and g(x)=f^(-1) (x). Which of the following describes the domain of g ?
A. x 4 D. x∈R
Answers
Answer:
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A
The domain of a composite function \displaystyle f\left(g\left(x\right)\right)f(g(x)) is the set of those inputs \displaystyle xx in the domain of \displaystyle gg for which \displaystyle g\left(x\right)g(x) is in the domain of \displaystyle ff.
HOW TO: GIVEN A FUNCTION COMPOSITION \DISPLAYSTYLE F\LEFT(G\LEFT(X\RIGHT)\RIGHT)F(G(X)), DETERMINE ITS DOMAIN.
Find the domain of g.
Find the domain of f.
Find those inputs, x, in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs, x, from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain of \displaystyle f\circ gf∘g.
EXAMPLE 8: FINDING THE DOMAIN OF A COMPOSITE FUNCTION
Find the domain of
\displaystyle \left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\frac{5}{x - 1}\text{ and }g\left(x\right)=\frac{4}{3x - 2}(f∘g)(x) wheref(x)=
x−1
5
and g(x)=
3x−2
4
SOLUTION
The domain of \displaystyle g\left(x\right)g(x) consists of all real numbers except \displaystyle x=\frac{2}{3}x=
3
2
, since that input value would cause us to divide by 0. Likewise, the domain of \displaystyle ff consists of all real numbers except 1. So we need to exclude from the domain of \displaystyle g\left(x\right)g(x) that value of \displaystyle xx for which \displaystyle g\left(x\right)=1g(x)=1.
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
4
3
x
−
2
=
1
4
=
3
x
−
2
6
=
3
x
x
=
2
So the domain of \displaystyle f\circ gf∘g is the set of all real numbers except \displaystyle \frac{2}{3}
3
2
and \displaystyle 22. This means that
\displaystyle x\ne \frac{2}{3}\text{or}x\ne 2x≠
3
2
orx≠2
We can write this in interval notation as
\displaystyle \left(-\infty ,\frac{2}{3}\right)\cup \left(\frac{2}{3},2\right)\cup \left(2,\infty \right)(−∞,
3
2
)∪(
3
2
,2)∪(2,∞)
EXAMPLE 9: FINDING THE DOMAIN OF A COMPOSITE FUNCTION INVOLVING RADICALS
Find the domain of
\displaystyle \left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\sqrt{x+2}\text{ and }g\left(x\right)=\sqrt{3-x}(f∘g)(x) wheref(x)=√
x+2
and g(x)=√
3−x
SOLUTION
Because we cannot take the square root of a negative number, the domain of \displaystyle gg is \displaystyle \left(-\infty ,3\right](−∞,3]. Now we check the domain of the composite function
\displaystyle \left(f\circ g\right)\left(x\right)=\sqrt{3-x+2}\text{ or}\left(f\circ g\right)\left(x\right)=\sqrt{5-x}(f∘g)(x)=√
3−x+2
or(f∘g)(x)=√
5−x
The domain of this function is \displaystyle \left(-\infty ,5\right](−∞,5]. To find the domain of \displaystyle f\circ gf∘g, we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since \displaystyle \left(-\infty ,3\right](−∞,3] is a proper subset of the domain of \displaystyle f\circ gf∘g. This means the domain of \displaystyle f\circ gf∘g is the same as the domain of \displaystyle gg, namely, \displaystyle \left(-\infty ,3\right](−∞,3].