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+10
g(x)=−8x−3
1. (f∘f)(x)=f(f(x))
We first find f(f(x)), which means substituting f(x) into f.
f(x)=7x+10, so substitute f(x) into f:
f(f(x))=f(7x+10)=7(7x+10)+10=49x+70+10=49x+80
Thus, (f∘f)(x)=49x+80.
2. (f∘g)(x)=f(g(x))
We now find f(g(x)), meaning substituting g(x) into f.
g(x)=−8x−3, so substitute g(x) into f:
f(g(x))=f(−8x−3)=7(−8x−3)+10=−56x−21+10=−56x−11
Thus, (f∘g)(x)=−56x−11.
3. (g∘f)(x)=g(f(x))
Next, find g(f(x)), meaning substituting f(x) into g.
f(x)=7x+10, so substitute f(x) into g:
g(f(x))=g(7x+10)=−8(7x+10)−3=−56x−80−3=−56x−83
Thus, (g∘f)(x)=−56x−83.
4. (g∘g)(x)=g(g(x))
Finally, find g(g(x)), meaning substituting g(x) into g.
g(x)=−8x−3, so substitute g(x) into g:
g(g(x))=g(−8x−3)=−8(−8x−3)−3=64x+24−3=64x+21
Thus, (g∘g)(x)=64x+21.
Final Answers:
(f∘f)(x)=49x+80
(f∘g)(x)=−56x−11
(g∘f)(x)=−56x−83
(g∘g)(x)=64x+21
Would you like further explanation or have any other questions?
Here are 5 related questions for practice:
What is (f∘g)(2) for the given functions f(x) and g(x)?
Find (g∘g)(−1) for the given function g(x).
If h(x)=2x2+3x, what is (f∘h)(x)?
What is the inverse of the function f(x)=7x+10?
Can you find (g∘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