Math Problem Statement

For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (a) f(x) = (x - 7)/2, g(x) = 2x + 7; (b) f(x) = -6x, g(x) = -x/6.

Solution

To solve this problem, let’s analyze the pairs of functions f(x)f(x) and g(x)g(x), compute the compositions f(g(x))f(g(x)) and g(f(x))g(f(x)), and determine whether they are inverses by verifying if both compositions equal xx.


Problem (a):

  • f(x)=x72f(x) = \frac{x - 7}{2}
  • g(x)=2x+7g(x) = 2x + 7

Step 1: Compute f(g(x))f(g(x))

Substitute g(x)=2x+7g(x) = 2x + 7 into f(x)f(x): f(g(x))=f(2x+7)=(2x+7)72f(g(x)) = f(2x + 7) = \frac{(2x + 7) - 7}{2} Simplify: f(g(x))=2x2=xf(g(x)) = \frac{2x}{2} = x

Step 2: Compute g(f(x))g(f(x))

Substitute f(x)=x72f(x) = \frac{x - 7}{2} into g(x)g(x): g(f(x))=g(x72)=2(x72)+7g(f(x)) = g\left(\frac{x - 7}{2}\right) = 2\left(\frac{x - 7}{2}\right) + 7 Simplify: g(f(x))=(x7)+7=xg(f(x)) = (x - 7) + 7 = x

Conclusion for (a):

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


Problem (b):

  • f(x)=6xf(x) = -6x
  • g(x)=x6g(x) = -\frac{x}{6}

Step 1: Compute f(g(x))f(g(x))

Substitute g(x)=x6g(x) = -\frac{x}{6} into f(x)f(x): f(g(x))=f(x6)=6(x6)f(g(x)) = f\left(-\frac{x}{6}\right) = -6\left(-\frac{x}{6}\right) Simplify: f(g(x))=xf(g(x)) = x

Step 2: Compute g(f(x))g(f(x))

Substitute f(x)=6xf(x) = -6x into g(x)g(x): g(f(x))=g(6x)=6x6g(f(x)) = g(-6x) = -\frac{-6x}{6} Simplify: g(f(x))=xg(f(x)) = x

Conclusion for (b):

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


Final Answers:

  1. (a): f(x)f(x) and g(x)g(x) are inverses of each other.
  2. (b): f(x)f(x) and g(x)g(x) are inverses of each other.

Would you like further clarification on any step? Let me know!


Related Questions:

  1. What does it mean for two functions to be inverses of each other?
  2. How do you find the inverse of a function algebraically?
  3. Can you provide an example of two functions that are not inverses?
  4. What happens if f(g(x))=xf(g(x)) = x but g(f(x))xg(f(x)) \neq x?
  5. How does domain and range impact whether functions can be inverses?

Tip:

When verifying inverses, always calculate both f(g(x))f(g(x)) and g(f(x))g(f(x)) to confirm they equal xx.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Inverse Functions
Simplifying Algebraic Expressions

Formulas

f(g(x))
g(f(x))

Theorems

Definition of Inverse Functions: f(g(x)) = x and g(f(x)) = x

Suitable Grade Level

Grades 10-12