advanced functions/precalculus

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)

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asked by Diane
  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)

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    posted by Reiny

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