What You Need to Know About Composite Functions

In mathematics, a function refers to a regulation that associates an offered collection of inputs to a set of possible returns. Composite functions depend on any other functions by composing one function within another function.

If given f[g(x)] as a function, that means that it’s the composite function of f(x) and g(x). the f(x) is the outer function of the composite function, and g(x) is the inner function. This composite function can also be read as the function g is the inner function of the outer function f.

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Start learning composite functions effectively with examples

To learn composite functions effectively, you must do a lot of practice by undertaking various examples offered in different math sites. Math is a subject that involves learning by doing, and that means the more you practice, the better you become. If possible, you can practice composite functions examples and solutions are available on Plainmath. YouTube is also another great resource that you can use to learn composite functions. 

The domain of composite functions

You can find the domain of a composite function using a graph. Also, you can identify the input values; if there is an even root, make sure that you exclude any real numbers that result in a negative value in the radicand. After that, set the radicand greater than or equal to zero and solve for the value of x.

That is to mean if f(x) = 4/ 2x-5

That is equating; 2x-5 = 0 and solving for the value of x

Taking 5 to the right-hand side of the equation changes its sign from negative to positive. i.e., 2x = 5 

Divide both sides by 2 to get the value of x as;

X = 5/2

For domain: all real values except x = 5/2

And the final answer is;

(-∞, 5/2) υ (5/2, ∞)

Forming composite functions using algebra

You can use various operations to form composite functions using algebra. One of the main reasons why math is important in every aspect of life is that you can play around with these operations to get your desired result. Through addition, subtraction, multiplication and division, you can combine different functions with a composite function. 

Suppose you have two columns of different numbers that represent two separate identities. Let's assume one of the columns represents the contribution by teachers in a school and the other represents students' contribution in that school towards a particular event organized annually.

And the results are the total contributions by the school. If g(x) is the teacher's contribution and h(x) contribution by the students. If we opt to use T to represent the total contribution by the school, then our new function is;

T(x) = g(x) + h(x)

If this holds a true value for every year the event is organized, then the relationship between the functions can be written as; T = g + h … without reference to the year.

Just as for the above example, you can define the difference, ratio of functions and product for any pair of functions that have the same inputs and the same kind of outputs. That doesn't mean they have to be numbered.

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How to solve composite functions

Solving composite functions entails finding the composition of two functions. In most cases, a small circle (∘) is used for the composition of the function. Here are the steps followed when solving a composite function.

i.e., If you have been given (f ∘ g) (x) = f [g (x)], you can rewrite it as (f ∘ g) (x²) = f [g (x²)]

For example, given the function; h (x) = x2 + 6 and g (x) = 2x – 1, find (h ∘ g) (x).

Solution

Substitute x with 2x – 1 in the function h(x) = x2 + 6.

(h ∘ g) (x) = (2x – 1)2 + 6 = (2x – 1) (2x – 1) + 6

Apply foil 

= 4x2 – 4x + 1 + 6

= 4x2 – 4x + 7

Conclusion 

Composite functions are applicable in most scenarios of life. Every time you are doing two computations, that's a composition of functions. On the other hand, a function is just one process that turns one thing into another. Any moment you are trying to solve an algorithm, you have a list of functions that you need to compose to make it valid. 

Author’s Bio 

Alisia Stren worked in the media sector for a long time before turning to freelance blogging and academic writing and editing. Her strong background of working with top companies has helped her to deliver her best work in tight deadlines on a consistent basis. Her free time is spent watching comedy shows, reading romantic novels and doing pencil sketching.