Math Problem Statement

Consider the functions f(x) = 7x + 10 and g(x) = -8x - 3. Determine the following compositions of functions: (f o f)(x), (f o g)(x), (g o f)(x), and (g o g)(x).

Solution

Let's solve the composition of functions problem displayed in the image. The given functions are:

  • f(x)=7x+10f(x) = 7x + 10
  • g(x)=8x3g(x) = -8x - 3

1. (ff)(x)=f(f(x))(f \circ f)(x) = f(f(x))

We first find f(f(x))f(f(x)), which means substituting f(x)f(x) into ff.

  • f(x)=7x+10f(x) = 7x + 10, so substitute f(x)f(x) into ff:
  • f(f(x))=f(7x+10)=7(7x+10)+10=49x+70+10=49x+80f(f(x)) = f(7x + 10) = 7(7x + 10) + 10 = 49x + 70 + 10 = 49x + 80

Thus, (ff)(x)=49x+80(f \circ f)(x) = 49x + 80.

2. (fg)(x)=f(g(x))(f \circ g)(x) = f(g(x))

We now find f(g(x))f(g(x)), meaning substituting g(x)g(x) into ff.

  • g(x)=8x3g(x) = -8x - 3, so substitute g(x)g(x) into ff:
  • f(g(x))=f(8x3)=7(8x3)+10=56x21+10=56x11f(g(x)) = f(-8x - 3) = 7(-8x - 3) + 10 = -56x - 21 + 10 = -56x - 11

Thus, (fg)(x)=56x11(f \circ g)(x) = -56x - 11.

3. (gf)(x)=g(f(x))(g \circ f)(x) = g(f(x))

Next, find g(f(x))g(f(x)), meaning substituting f(x)f(x) into gg.

  • f(x)=7x+10f(x) = 7x + 10, so substitute f(x)f(x) into gg:
  • g(f(x))=g(7x+10)=8(7x+10)3=56x803=56x83g(f(x)) = g(7x + 10) = -8(7x + 10) - 3 = -56x - 80 - 3 = -56x - 83

Thus, (gf)(x)=56x83(g \circ f)(x) = -56x - 83.

4. (gg)(x)=g(g(x))(g \circ g)(x) = g(g(x))

Finally, find g(g(x))g(g(x)), meaning substituting g(x)g(x) into gg.

  • g(x)=8x3g(x) = -8x - 3, so substitute g(x)g(x) into gg:
  • g(g(x))=g(8x3)=8(8x3)3=64x+243=64x+21g(g(x)) = g(-8x - 3) = -8(-8x - 3) - 3 = 64x + 24 - 3 = 64x + 21

Thus, (gg)(x)=64x+21(g \circ g)(x) = 64x + 21.

Final Answers:

  1. (ff)(x)=49x+80(f \circ f)(x) = 49x + 80
  2. (fg)(x)=56x11(f \circ g)(x) = -56x - 11
  3. (gf)(x)=56x83(g \circ f)(x) = -56x - 83
  4. (gg)(x)=64x+21(g \circ g)(x) = 64x + 21

Would you like further explanation or have any other questions?

Here are 5 related questions for practice:

  1. What is (fg)(2)(f \circ g)(2) for the given functions f(x)f(x) and g(x)g(x)?
  2. Find (gg)(1)(g \circ g)(-1) for the given function g(x)g(x).
  3. If h(x)=2x2+3xh(x) = 2x^2 + 3x, what is (fh)(x)(f \circ h)(x)?
  4. What is the inverse of the function f(x)=7x+10f(x) = 7x + 10?
  5. Can you find (gf)(0)(g \circ f)(0) using the given functions?

Tip: When composing functions, always substitute the entire expression of the inner function into the outer function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Linear Functions
Substitution

Formulas

f(x) = 7x + 10
g(x) = -8x - 3

Theorems

Function Composition

Suitable Grade Level

Grades 9-12