Math Problem Statement
If
f
(
x
)
x
+
9
and
g
(
x
)
x
−
9
,
(a)
f
(
g
(
x
)
)
(b)
g
(
f
(
x
)
)
(c) Thus
g
(
x
)
is called an
function of
f
(
x
)
Solution
Given the functions and , let's solve the parts step by step.
(a) Find
This means we substitute into :
Since , we replace in with :
So, .
(b) Find
This means we substitute into :
Since , we replace in with :
So, .
(c) Thus, is called an inverse function of
Since both and , the functions and are inverses of each other.
Let me know if you'd like further details or have any questions!
Here are five related questions:
- What are the properties of inverse functions?
- How do you prove that two functions are inverses of each other?
- Can all functions have inverses? If not, why?
- How do the graphs of inverse functions relate to each other?
- How does the composition of functions work in general?
Tip: If and , then and are inverse functions. Always check both compositions!
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Math Problem Analysis
Mathematical Concepts
Algebra
Function Composition
Inverse Functions
Formulas
f(x) = x + 9
g(x) = x - 9
f(g(x)) = f(x - 9) = x
g(f(x)) = g(x + 9) = x
Theorems
Inverse Function Theorem
Suitable Grade Level
Grades 9-12
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