Math Problem Statement

  1. If f(x) = 3x + 6 and g(x) = 𝑥3 + 3 then g[f(x)]= A) (5-x) B) (x-5) C)(x+3) D) (x+5)
  2. If f(x) = 4x – 1 and g(x) = 𝑥4 + 14 then f[g(x)]= A) (3-x) B) (x-3) C) x D) (x-1) اختر بس

Solution

Let's solve each function composition one at a time.

14. For g[f(x)]g[f(x)]

  1. Find f(x)f(x): f(x)=3x+6f(x) = 3x + 6

  2. Substitute f(x)f(x) into g(x)g(x): g[f(x)]=g(3x+6)=(3x+6)3+3g[f(x)] = g(3x + 6) = (3x + 6)^3 + 3 This expression is complex, but we can analyze the answer choices instead.

    • After substitution and simplification, you can check values for xx to find which answer matches.

    Evaluating the original functions with a specific xx (e.g., x=0x = 0): f(0)=6    g(6)=63+3=216+3=219f(0) = 6 \implies g(6) = 6^3 + 3 = 216 + 3 = 219

    None of the options fit directly, but checking specific values or further simplifying might help find the right form.

15. For f[g(x)]f[g(x)]

  1. Find g(x)g(x): g(x)=x4+14g(x) = x^4 + 14

  2. Substitute g(x)g(x) into f(x)f(x): f[g(x)]=f(x4+14)=4(x4+14)1=4x4+561=4x4+55f[g(x)] = f(x^4 + 14) = 4(x^4 + 14) - 1 = 4x^4 + 56 - 1 = 4x^4 + 55 Again, check values for xx to find which matches.

Answers

  1. For the first function g[f(x)]g[f(x)], further simplification or substitution is required to find an exact match.
  2. For the second function f[g(x)]f[g(x)], it simplifies to 4x4+554x^4 + 55, which doesn’t match given options.

It seems neither question has a straightforward answer from the provided options without further evaluation.

Would you like further details on these calculations or help with any specific part? Here are some related questions:

  1. How do you evaluate a function at a specific point?
  2. What is the process of function composition?
  3. Can you explain how to simplify polynomial expressions?
  4. What techniques are used to analyze complex function compositions?
  5. How do you approach finding roots of polynomial functions?

Tip: When dealing with compositions, substituting specific values can sometimes clarify the function's behavior without needing full simplification.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Functions

Formulas

f(x) = 3x + 6
g(x) = x^3 + 3
f(x) = 4x - 1
g(x) = x^4 + 14

Theorems

Composition of Functions

Suitable Grade Level

Grades 9-11