Math Problem Statement

Let f(x) = 5x + 1 and g(x) = x^2 - 1. Find the value of (f ∘ g)(2) and (g ∘ f)(1/6).

Solution

Let's solve each question based on the information provided.

Question 9

Given:

  • f(x)=5x+1f(x) = 5x + 1
  • g(x)=x21g(x) = x^2 - 1

We need to find (fg)(2)(f \circ g)(2), which means evaluating f(g(2))f(g(2)).

Steps:

  1. Calculate g(2)g(2): g(2)=221=41=3g(2) = 2^2 - 1 = 4 - 1 = 3

  2. Substitute g(2)g(2) into f(x)f(x): f(g(2))=f(3)=53+1=15+1=16f(g(2)) = f(3) = 5 \cdot 3 + 1 = 15 + 1 = 16

So, (fg)(2)=16(f \circ g)(2) = 16.

Question 10

Given:

  • f(x)=2x+1f(x) = 2x + 1
  • g(x)=x21g(x) = x^2 - 1

We need to find (gf)(16)(g \circ f)\left(\frac{1}{6}\right), which means evaluating g(f(16))g(f\left(\frac{1}{6}\right)).

Steps:

  1. Calculate f(16)f\left(\frac{1}{6}\right): f(16)=216+1=26+1=13+1=43f\left(\frac{1}{6}\right) = 2 \cdot \frac{1}{6} + 1 = \frac{2}{6} + 1 = \frac{1}{3} + 1 = \frac{4}{3}

  2. Substitute f(16)f\left(\frac{1}{6}\right) into g(x)g(x): g(43)=(43)21=1691=16999=79g\left(\frac{4}{3}\right) = \left(\frac{4}{3}\right)^2 - 1 = \frac{16}{9} - 1 = \frac{16}{9} - \frac{9}{9} = \frac{7}{9}

So, (gf)(16)=79(g \circ f)\left(\frac{1}{6}\right) = \frac{7}{9}.

If you need further details or have any questions, please let me know!


Here are 5 related questions to expand on this information:

  1. How would you find (fg)(x)(f \circ g)(x) in general terms, without a specific input?
  2. What is the domain of (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x) for the given functions ff and gg?
  3. Can function composition be commutative, i.e., is (fg)(x)=(gf)(x)(f \circ g)(x) = (g \circ f)(x) always true? Why or why not?
  4. What happens if we change the functions f(x)f(x) and g(x)g(x) to other types of functions (e.g., exponential or logarithmic)?
  5. How can we find the range of the composed function (fg)(x)(f \circ g)(x)?

Tip: When performing function composition, always evaluate the innermost function first before substituting into the outer function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Functions

Formulas

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

Theorems

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Suitable Grade Level

Grade 11