Math Problem Statement

If f(x) = 11 6x + 5 and g(x) =

2x + 7 , compute the compositions (f ∘ g)(1) and (g ∘ f)(1).

Solution

Let's compute the compositions (fg)(1)(f \circ g)(1) and (gf)(1)(g \circ f)(1) step by step, using the given functions:

  • f(x)=116x+5f(x) = \frac{11}{6x + 5}
  • g(x)=2x+7g(x) = 2x + 7

1. Compute (fg)(1)(f \circ g)(1)

The composition (fg)(1)(f \circ g)(1) means we first apply g(1)g(1), then apply ff to the result.

  1. Find g(1)g(1): g(1)=2(1)+7=2+7=9g(1) = 2(1) + 7 = 2 + 7 = 9
  2. Now apply ff to g(1)=9g(1) = 9: f(9)=116(9)+5=1154+5=1159f(9) = \frac{11}{6(9) + 5} = \frac{11}{54 + 5} = \frac{11}{59} Thus, (fg)(1)=1159(f \circ g)(1) = \frac{11}{59}.

2. Compute (gf)(1)(g \circ f)(1)

The composition (gf)(1)(g \circ f)(1) means we first apply f(1)f(1), then apply gg to the result.

  1. Find f(1)f(1): f(1)=116(1)+5=116+5=1111=1f(1) = \frac{11}{6(1) + 5} = \frac{11}{6 + 5} = \frac{11}{11} = 1
  2. Now apply gg to f(1)=1f(1) = 1: g(1)=2(1)+7=2+7=9g(1) = 2(1) + 7 = 2 + 7 = 9 Thus, (gf)(1)=9(g \circ f)(1) = 9.

Final Answer:

  • (fg)(1)=1159(f \circ g)(1) = \frac{11}{59}
  • (gf)(1)=9(g \circ f)(1) = 9

Would you like more details on any of the steps?

Here are 5 related questions:

  1. What is the general formula for (fg)(x)(f \circ g)(x)?
  2. How does the composition of functions differ from addition or multiplication of functions?
  3. Can we reverse the composition order for any two functions?
  4. What happens if we change the given functions f(x)f(x) and g(x)g(x)?
  5. What would (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x) look like as general expressions?

Tip: Always be cautious about the order of compositions; fgf \circ g is not the same as gfg \circ f.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra

Formulas

(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))

Theorems

Composition of Functions

Suitable Grade Level

Grades 10-12