Composite Functions: Understanding the Basics
Composite Functions – Introduction
In A Level Maths, composite functions play a significant role in understanding the relationships between different mathematical expressions. In this blog post, we will delve into the concept of composite functions and explore how they are used in mathematical analysis.
Defining Composite Functions
A composite function is formed by combining two or more functions, where the output of one function is used as the input for another. In simpler terms, it involves applying one function to the result of another function. This chaining of functions allows us to model complex relationships and study their behaviour.
Notation
The notation used to represent composite functions involves the composition symbol (∘) and parentheses. Suppose we have two functions, f(x) and g(x). The composite function of f and g can be denoted as f(g(x)) or (f ∘ g)(x). It is important to note that the order in which functions are composed matters, as changing the order will yield different results.
Evaluating Composite Functions
To evaluate a composite function, we substitute the output of one function into the input of another. For example, if we have f(x) = 2x + 1 and g(x) = x^2, we can evaluate the composite function f(g(x)) as follows:
Evaluate g(x): g(x) = x^2.
Substitute the result of g(x) into f(x): f(g(x)) = f(x^2).
Simplify f(x^2): f(g(x)) = 2(x^2) + 1 = 2x^2 + 1.
Properties of Composite Functions
Composite functions possess several properties that are worth noting:
Associativity: The order of function composition is associative, meaning that for functions f, g, and h, the composition can be written as (f ∘ g) ∘ h = f ∘ (g ∘ h). However, the results may differ based on the specific functions used.
Identity Function: The identity function serves as the identity element for function composition. For any function f, we have f ∘ I = I ∘ f = f, where I represents the identity function.
Inverse Functions: If f and g are inverse functions, then f(g(x)) = g(f(x)) = x. This property allows us to define the composition of a function and its inverse.
Applications of Composite Functions
Composite functions find applications in various areas of mathematics and real-world scenarios. Some notable examples include:
Modelling transformations in geometry, such as rotations, translations, and reflections.
Analysing complex systems, such as electrical circuits, where multiple functions interact.
Describing iterative processes, such as compound interest or population growth.
Composite Functions Example 1
Solution
Part a)
\begin{aligned} f g(x) & =f\left(x^2+4\right) \\ & =3\left(x^2+4\right)+2 \\ & =3 x^2+14 . \end{aligned}Part b)
\begin{aligned} g f(x) & =g(3 x+2) \\ & =(3 x+2)^2+4 . \\ & =9 x^2+12 x+8 \end{aligned}Part c)
\begin{aligned} f^2(x) & =f(3 x+2) \\ & =3(3 x+2)+2 \\ & =9 x+8 \end{aligned}Part d)
\begin{aligned} \mathrm{fg}(x) & =3 x^2+14 \\ \mathrm{fg}(b) & =62 \\ 3 b^2+14 & =62 \\ b^2 & =16 \\ b & = \pm 4 \end{aligned}Composite Functions Example 2
Solution
Part a)
\begin{aligned} \operatorname{gf}(x) & =\mathrm{g}\left(\mathrm{e}^{5 x}\right) \\ & =4 \ln \left(\mathrm{e}^{5 x}\right) \\ & =4(5 x) \\ & =20 x, x \in \mathbb{R} \end{aligned}Part b)
\begin{aligned} \mathrm{fg}(x) & =\mathrm{f}(4 \ln x) \\ & =\mathrm{e}^{5(4 \ln x)} \\ & =\mathrm{e}^{\ln x^{20}} \\ & =x^{20}, x \in \mathbb{R}, x>0 \end{aligned}Composite Functions Example 3
Solution
Part a)
\begin{aligned} \mathrm{fg}(x) & =\mathrm{f}[\mathrm{g}(x)] \\ & =\mathrm{f}\left(x^2\right) \\ & =x^2+1 \end{aligned}Part b)
\begin{aligned} \operatorname{gh}(x) & =\mathrm{g}[\mathrm{h}(x)] \\ & =\mathrm{g}(3 x) \\ & =(3 x)^2 \\ & =9 x^2 \end{aligned}Part c)
\begin{aligned} \operatorname{hgf}(x) & =\mathrm{hg}[\mathrm{f}(x)] \\ & =\mathrm{h}[\mathrm{g}(x+1)] \\ & =\mathrm{h}\left[(x+1)^2\right] \\ & =3(x+1)^2 \end{aligned}Part d)
\begin{aligned} \mathrm{f}^2(x) & =\mathrm{f}[\mathrm{f}(x)] \\ & =\mathrm{f}(x+1) \\ & =(x+1)+1 \\ & =x+2 \end{aligned}Conclusion
Composite functions provide a powerful tool for studying the relationships between different mathematical expressions. By combining functions, we can gain deeper insights into complex systems and model real-world phenomena. Understanding the basics of composite functions is essential for tackling more advanced mathematical concepts and their applications.