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Marta_Voda [28]
3 years ago
5

Given f: ℝ → ℝ and : ℝ → ℝ , for the following, find f(g(x)) and g(f(x)), and state the domain.

Mathematics
1 answer:
horrorfan [7]3 years ago
3 0

Answer:

Remember, (f\circ g)(x)=f(g(x)) and the range of g must be in the domain of f.

a)

f(g(x))=f(x-1)=(x-1)^2-(x-1)=x^2-2x+1-x+1=x^2-3x+2

g(f(x))=g(x^2-x)=(x^2-x)-1=x^2-x-1

The domain of f(g(x)) and g(f(x)) is the set of reals.

b)

f(g(x))=f(\sqrt{x}-2)=(\sqrt{x}-2)^2-x=\sqrt{x}^2-2*2*\sqrt{x}+2^2-x=-4\sqrt{x}+4

g(f(x))=g(x^2-x)=\sqrt{x^2-x}-2

The domain of f(g(x)) is the set of nonnegative reals and the domain of g(f(x)) is the set of number such that x^2-x\geq 0

c)

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

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

The domain of f(g(x)) is the set of reals except the 1 and the domain of g(f(x)) is the set of reals except the 1 and -1

d)

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

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

The domain of f(g(x)) is the set of reals except 2, and the domain of g(f(x)) is the set of reals except -1.

e)

f(g(x))=f(log(2(x+3)))=f(log(2x+6))=log(2x+6)-1

g(f(x))=g(x-1)=log(2(x-1)+6)=log(2x+4)

The domain of f(g(x)) is the set of nonnegative reals except -3. The domain of g(f(x)) is the set of nonnegative reals except -2.

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