Math Problem Statement

Misalkan fungsi f dirumuskan dengan f(x) = x + 2 dan g dirumuskan dengan g(x) = x^3. Temukan komposisi f(g(x)) dan g(f(x)).

Solution

The image you uploaded contains information on function definitions and compositions. Let's break down the given functions and solve the related questions:

  1. Function Definitions:

    • Function f(x)=x3f(x) = x^3
    • Using formula f(x)=x+1f(x) = x + 1, we have:
      • f(1)=1+1=2f(1) = 1 + 1 = 2
      • f(2)=2+1=3f(2) = 2 + 1 = 3
      • f(3)=3+1=4f(3) = 3 + 1 = 4
      • f(t)=t+1f(t) = t + 1 for a general tt.
  2. Function g(x)g(x) and Composition:

    • The composition f(g(x))f(g(x)) results in g(x)+1g(x) + 1.
    • If h(x)=f(g(x))h(x) = f(g(x)), then h(x)=g(x)+1h(x) = g(x) + 1.
  3. Computing g(f(x))g(f(x)):

    • Given the composition in the reverse order, g(f(x))=f(x)+1g(f(x)) = f(x) + 1 since g(x)g(x) is expected to behave similarly to f(x)f(x) based on its usage in h(x)h(x).

Without the explicit definition for g(x)g(x), we assume it might take the form g(x)=xg(x) = x based on the pattern observed, meaning g(x)=xg(x) = x just returns xx, making g(f(x))=f(x)g(f(x)) = f(x). However, this is based on the general pattern shown and could differ with more information about g(x)g(x).

Would you like more detailed explanations on any of these parts or help with another function composition?

Here are some related questions to consider:

  1. What would be the result of f(f(x))f(f(x))?
  2. How would g(g(x))g(g(x)) be calculated if g(x)g(x) had a different definition, such as g(x)=x+2g(x) = x + 2?
  3. What is the graphical interpretation of composing functions like ff and gg?
  4. Could you explore the properties of the identity function within the context of function compositions?
  5. How do inverses of functions relate to function compositions?

Tip: When working with function compositions, always verify the inner function's output domain matches the outer function's input domain.

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(x) = x + 2
g(x) = x^3

Theorems

Function Composition Rule

Suitable Grade Level

Grades 10-12