f(x)=2x+5 g(x)=x^2-3 h(x)=7-x/3

Question

Compute (f-h)(4)

Evaluate the following compositions: (f o g)(x) and (h o g)(x)

Find the inverse functions: f^-1(x) and h^-1(x)

Answer 1

(f-h)(x)=2x+5-7+x/3=7x/3-2, (f-h)(4)=28/3-2=22/3.

(f ○ g)(x)=f(g(x))=2g(x)+5=2(x²-3)+5=2x²-1.

(h ○ g)(x)=h(g(x))=7-g(x)/3=7-(x²-3)/3=7-x²/3+1=8-x²/3.

Let y=f(x)=2x+5, 2x=y-5, x=(y-5)/2=F(y), so F(x)=(x-5)/2. But F(x)=f^-1(x), so f^-1​(x)=(x-5)/2.

(f ○ f^-1)(x)=f(f^-1(x))=2f^-1(x)+5=2[(x-5)/2]+5=x (which is the definition of the inverse).

Similarly, let y=h(x)=7-x/3, 3y=21-x, x=21-3y=H(y), so h^-1(x)=H(x)=21-3x.