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 f(x)=x+9f(x) = x + 9 and g(x)=x9g(x) = x - 9, let's solve the parts step by step.

(a) Find f(g(x))f(g(x))

This means we substitute g(x)g(x) into f(x)f(x):

f(g(x))=f(x9)f(g(x)) = f(x - 9)

Since f(x)=x+9f(x) = x + 9, we replace xx in f(x)f(x) with g(x)=x9g(x) = x - 9:

f(g(x))=(x9)+9=xf(g(x)) = (x - 9) + 9 = x

So, f(g(x))=xf(g(x)) = x.

(b) Find g(f(x))g(f(x))

This means we substitute f(x)f(x) into g(x)g(x):

g(f(x))=g(x+9)g(f(x)) = g(x + 9)

Since g(x)=x9g(x) = x - 9, we replace xx in g(x)g(x) with f(x)=x+9f(x) = x + 9:

g(f(x))=(x+9)9=xg(f(x)) = (x + 9) - 9 = x

So, g(f(x))=xg(f(x)) = x.

(c) Thus, g(x)g(x) is called an inverse function of f(x)f(x)

Since both f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x, the functions f(x)f(x) and g(x)g(x) are inverses of each other.


Let me know if you'd like further details or have any questions!

Here are five related questions:

  1. What are the properties of inverse functions?
  2. How do you prove that two functions are inverses of each other?
  3. Can all functions have inverses? If not, why?
  4. How do the graphs of inverse functions relate to each other?
  5. How does the composition of functions work in general?

Tip: If f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x, then ff and gg 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