Question

1. Given f(x)=x^2 and g(x)=x+2, determine in standard form:
a)g(f(x))
b)when f(g(x))=g(f(x))

for #1 would the answer for a be x^2+2?

2. Given the functions f(x)=5x+7 and g(x)=√x state the domain of g(x) and g(f(x))

3. Why is point (a,a) a point on f-1(f(x)) if (a,b) is a point on f(x)?

(f-1 means inverse)

Answer 1

f(g(x)
= f(x+2)
= (x+2)^2 or x^2 + 4x + 4

2.

if g(x) = √x , domain of g(x) is all x ≥ 0
g(f(x))
= g(5x+7)
= √(5x+7)
whose domain is :
5x+7 ≥ 0
5x ≥ -7
x ≥ -7/5

3. by definition
f^-1 (f(x)) = x

so if x = a
f^-1 (f(a)) = a

let's illustrate with an example
let f(x) = 2x+5
then f^-1 (x) = (x-5)/2
let x=3
f(3) = 11 , so (3,11) is on f(x)

then f^-1(f(11)) = (11-5)/2 = 3
giving us (3,3)