For All Values Of X(2)f(x) = (x + 2)2 And G(x) = 3(x - 1)a) Find Gf(x), Giving Your Answer In The Form
Understanding the composition of functions is a fundamental concept in algebra and calculus, especially when dealing with complex expressions and transformations. In this article, we will explore how to find the composite function Gf(x) given specific functions f(x) and G(x). The problem states that for all values of x, f(x) = (x + 2)^2 and G(x) = 3(x - 1). Our goal is to compute Gf(x), which means applying G to the output of f(x), and express the result in a simplified, standard form.
This process involves a step-by-step substitution and algebraic manipulation, which not only reinforces understanding of function composition but also enhances algebraic skills. Whether you are a student preparing for exams or someone interested in mathematical concepts, mastering this technique will deepen your comprehension of how functions interact.
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Understanding Function Composition
Before diving into calculations, it is essential to understand what function composition entails.
What Is Function Composition?
Function composition involves combining two functions such that the output of one function becomes the input of another. If f and G are functions, then:
- The composition of G after f is written as G∘f or Gf(x), which equals G(f(x))
- Similarly, the composition of f after G is written as f∘G or fG(x), which equals f(G(x))
In this context, since we are asked to find Gf(x), it means:
Gf(x) = G(f(x))
Why Is Function Composition Important?
Function composition is crucial in various fields such as calculus, computer science, physics, and engineering because it models layered processes or transformations. It helps analyze how a change in the input propagates through multiple functions, ultimately impacting the output.
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Given Functions and What Is Being Asked
The problem provides two functions:
- f(x) = (x + 2)^2
- G(x) = 3(x - 1)
and asks to find Gf(x), which is G applied to f(x).
Objective:
Calculate G(f(x)) and express the answer in a simplified algebraic form.
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Step-by-Step Solution to Find Gf(x)
To find Gf(x), follow these systematic steps:
Step 1: Write Down the Composition
By definition:
Gf(x) = G(f(x))
which means replace every occurrence of x in G(x) with f(x).
Step 2: Substitute f(x) into G(x)
Given G(x) = 3(x - 1), replace x with f(x):
Gf(x) = 3(f(x) - 1)
Since f(x) = (x + 2)^2, substitute:
Gf(x) = 3(((x + 2)^2) - 1)
Step 3: Simplify the Expression Inside the Parentheses
Focus on simplifying:
((x + 2)^2) - 1
Recall that (x + 2)^2 expands to:
x^2 + 4x + 4
Thus,
((x + 2)^2) - 1 = (x^2 + 4x + 4) - 1 = x^2 + 4x + 3
Step 4: Multiply by 3
Now, multiply the entire expression by 3:
Gf(x) = 3(x^2 + 4x + 3)
Distribute the 3 across all terms:
Gf(x) = 3x^2 + 12x + 9
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Final Expression and Its Significance
The simplified form of Gf(x) is:
Gf(x) = 3x^2 + 12x + 9
This quadratic expression clearly shows how the composition of the given functions results in a polynomial.
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Expressing Gf(x) in the Requested Form
The problem asks to give the answer in a specific form, which, although not explicitly provided in the excerpt, typically refers to standard polynomial form or factored form.
Standard Polynomial Form:
Gf(x) = 3x^2 + 12x + 9
Factored Form:
We can factor Gf(x) further:
First, factor out the common factor 3:
Gf(x) = 3(x^2 + 4x + 3)
Next, factor the quadratic inside the parentheses:
x^2 + 4x + 3 = (x + 1)(x + 3)
Therefore:
Gf(x) = 3(x + 1)(x + 3)
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Conclusion and Applications
By carefully substituting and simplifying, we have determined that:
Gf(x) = 3(x + 1)(x + 3)
This process illustrates the importance of understanding function composition and algebraic manipulation. It also emphasizes the value of recognizing factorizable quadratics to express functions in their most simplified, factored form.
Applications of this concept include:
- Solving complex algebraic problems
- Analyzing transformations in coordinate geometry
- Understanding layered functions in calculus
- Modeling real-world situations involving multiple variables
Mastering these techniques enhances mathematical fluency and prepares learners for more advanced topics such as derivatives, integrals, and functional analysis.
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Additional Tips for Mastering Function Composition
- Always clearly write down the functions involved before starting calculations.
- Substitute step-by-step, avoiding shortcuts that could lead to mistakes.
- Simplify expressions systematically, using algebraic identities where applicable.
- Practice with different types of functions (linear, quadratic, polynomial, exponential) to build confidence.
In summary, combining functions like f(x) = (x + 2)^2 and G(x) = 3(x - 1) involves substitution and algebraic simplification. The resulting composite function Gf(x) = 3(x + 1)(x + 3) demonstrates the layered nature of functions and the importance of careful algebraic manipulation. This fundamental skill is essential for tackling more complex problems across mathematics and related disciplines.
