Question

Let f(x) = x^2 and g(x) = 2x + 3. Write an expression that represents each composition:
a. (g ∘ f)(x)
b. f(f(−2))
c. (f ∘ g) (1 / x)

Answer 1

Answer:

Remember, if f and g are functions then $(f\circ g)(x)=f(g(x))$, then:

a)$(g\circ f)(x)=g(f(x)))=g(x^2)=2(x^2)+3=2x^2+3$

b) $f(f(-2))=f((-2)^2)=f(4)=4^2=16$

c)

$(f \circ g)(\frac{1}{x})=f(g(\frac{1}{x}))\\=f(2(\frac{1}{x})+3)\\=f(\frac{2}{x}+3)\\=(\frac{2}{x}+3)^2\\ =(\frac{2}{x})^2+2*\frac{2}{x}*3+3^2 \\ =\frac{4}{x^2}+\frac{12}{x}+9$

then, $(f\circ g)=\frac{4}{x^2}+\frac{12}{x}+9$