Question

Find the indicated function given f(x)=x^2-1 and g(x)=x+4. When typing your answer if you have an exponent then use the carrot key ^ by pressing SHIFT and 6. Type your simplified answers in descending powers of x an do not include any spaces between your characters.f(g(2))=Answerf(g(x))=Answerg(f(x))=Answer (g \circ g)(x) =Answer (f \circ f)(-1) =Answer

Answer 1

1. 35
2. f(g(x)) = x^2+8x+15
3. x^2+3
4. x+8
5. -1

Explanation:

You want various compositions and their values of the functions ...

• f(x) = x^2-1
• g(x) = x+4

1. f(g(2))

These are evaluated according to the order of operations. This means you start with the inner parentheses.

f(g(2) = f(2+4) = f(6) = 6^2-1 = 36-1

f(g(2)) = 35

2. f(g(x))

Same deal, using x instead of 2.

f(g(x) = f(x+4)

f(g(x)) = (x+4)^2-1 = x^2+8x+16-1

f(g(x)) = x^2+8x+15

3. g(f(x))

g(f(x)) = g(x^2-1) = (x^2-1)+4

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

4. g(g(x))

g(g(x)) = g(x+4) = (x+4)+4

g(g(x)) = x+8

5. f(f(-1))

f(f(-1)) = f((-1)^2-1) = f(1-1) = f(0) = 0^2-1

f(f(-1)) = -1

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Additional comment

In the attached calculator results, we defined Y0(x) = x^2-1, and Y1(x) = x+4, corresponding to f(x) and g(x).

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