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Thecombination of functionsis an important and fundamental part of algebraic mathematics. It is a skill that is applied across nearly all areas where mathematics is prevalent and is an indispensable concept in describing relationships between functions mathematically.
We can create a new function by adding, subtracting, multiplying, or dividing functions, just like how these operations form a new number.
The combination of functions is the act of combining multiple functions into a single function. This usually entails using basic mathematical operators such as addition or multiplication.
Functions can be combined using the same basic operators as any standard numerical set. These operators are:
As in other areas of math, the result of addition is thesum, the result of subtraction is thedifference, the result of the multiplication is theproductand the result of division is thequotient.
There are simple notation rules to follow when combining functions using these operators.
Addition
Subtraction
Multiplication
Division
Remember, we can only perform division when the denominator (the bottom of the fraction) is not zero. Why? Because dividing by zero is impossible!
There is another way that functions can be combined, known as theComposition of Functions. This entails forming whole a new function by using one function as the input to another. For instance,
Confused? Well, not to worry, we have an explanation specially dedicated just to that topic!
Consider the functions
and
.
What happens when we combine them by addition?
Now we can substitute each of the functions.
And finally, collect like terms.
Let's consider the same functions again,
and
.
What happens when we combine them by subtraction?
Now we substitute the functions as before.
And we can collect like terms.
Let's consider the functions
and
What happens if we multiply them together.
And substitute in the functions.
We then multiply out the brackets.
And again we can collect like terms.
Let's consider the functions
and
once again.
What happens when we combine them by division.
And substitute in the functions.
Then we simplify algebraically. In the case of these functions, we find a common factor and divide through.
Let's take a look at a practical example of how functions get combined.
(1)
Below is a composite shape, a circle contained within a square. Each side of the square is a tangent to the circle. Find a function for the total shaded area of the shape in terms of the circle radiusr.
A square with the area of a circle taken out of its centre, StudySmarter Originals
Solution:
Firstly, we must find the functions for the areas of the circle and the square.
We know the equation for the area of a circle is
We also can deduce that, if the length of each side of this square is the diameter of the circle, then the equation for the area of the square in terms of r is
We can define these as functions
Now, to find the function of the shaded area, we simply subtract the function for the area of the circle from the area of the square.
And then subbing in the functions we get
And once we simplify by expanding brackets and collecting like terms, we are left with a neat equation for the shaded area in terms ofrincm.
(2)
A store has a sale meaning that all clothes are
off. You wish to buy a pair of jeans that originally cost
. The sales tax is
. By constructing a composite function, find the final price of the jeans.
Solution:
To construct the composite function, we first must construct each constituent function. The first function,
, will be the function for the price of the jeans after the discount, and the second function,
, will be for the total price of the jeans.
Using
as the input to
我们得到的复合函数
Where
is the original cost of the jeans. Therefore the total cost of the jeans will be
Graphing combinations of functions can be done simply in two steps:
Combine the relevant functions into a single function.
Graph the newly created function.
What is really interesting, however, is seeing how combining two functions will alter the original graphs of those functions. For instance, if we combined the functions
and
, how do you think the graph of our combined function would compare to the graph of
Let's work through it and find out.
First, let's simply combine the functions
and
into a single function
by addition.
And next as always we substitute in our functions
and
.
And as before, we simply gather like terms to find our final combined function.
Plotting the two original functions on a graph with the new combined function gives us the plot below. What can we notice about how
relates to
and
Well, as we can see,
has a greater gradient than either
or
In fact, the gradient of
is equal to the gradients of
and
combined!
When we consider what we are really doing in combining functions, this makes perfect sense. The output of a combined function (the number on theverticalaxis) for a given input (the number on thehorizontalaxis) is simply a combination of the outputs of each original function.
To test this, pick a value on thex-axis,and find they-axisvalue for each of the functions. Does they-axisvalue for
equal they-axisvalues for
and
added together?
, in just the same way as numbers can.One way to combine functions is by addition,h(x) = g(x) + f(x).
Adding or subtracting linear functions from one another will always result in a linear function.
Combination of functions is a set of rules for merging multiple functions into one function in specific ways.
To find the combination of two functions, simply write a third function that combines the original two functions in the desired way, for instance,h(x) = f(x) - g(x),and substitute each of the original functions into the equation. From there it is just a case of tidying up the new combined function to get its simplest form.
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