In this explainer, we will learn how to define and evaluate functions and find the roots and the maximum or minimum of quadratic functions.
function is a mathematical relationship that takes inputs and maps each input onto exactly one output. Functions can be defined over many different objects; however, in this explainer, we will only focus on functions over sets of numbers.
It is important to note that when we input a value into a function, we always get the same output; we can never get a different output value. We can represent functions using function notation. If is the input value to a function , then we can represent the output of this function at as . There are many ways of representing functions, including an expression, a diagram, or a rule.
Definition: Function
A function is mathematical relationship that takes inputs and maps each input onto exactly one output.
For example, the function might double a value. Then, when we input 5, the function will double this value to output 10. We can describe this function using a rule and express the inputs and corresponding outputs using function notation. If we say that the input is , then the output would be . We can write this function as , where is the name of the function, is the input, and is the output.
We can also represent functions using diagrams. For instance, letβs say that we have a function that can take inputs of 1, 2, and 3 and the function adds 2 to each value. We can calculate the outputs of the function as 3, 4, and 5. We show this by connecting the inputs to the outputs using an arrow.
Of course, we could extend this function to include more input values; however, we are told that the only valid inputs are 1,2, and 3. In general, we say that the domain is the set of all valid input values.
For example, consider the function . We can evaluate the function by substituting values of into the function. If we let , then we see that
Thus, 2 is a possible input for the function. However, we cannot input every number. If we tried to input using the same method, we would divide by zero. This is not allowed, so we can conclude that 0 is not a valid input for this function. We call the set of all input values the domain of the function. So, the domain of this function is .
In our example of , we can input any real value of , so we can say that its domain is .
Letβs now consider the outputs of the function . Is any output possible? Let be any real number. We want to find an input value of so that the function outputs . Inputting into the function gives
Thus, any real value is a possible output of the function. The set of all output values of a function is called its range. So, the range of the function is . We can summarize these definitions as follows.
Definition: Domain and Range of a Function
The set of all possible input values of a function is called its domain.
The set of all possible output values of a function is called its range.
Letβs now see an example of finding the sum of two functions evaluated at different values.
Example 1: Finding the Value of the Addition of Two Functions at Given Values
Given that and , find the value of .
Answer
We evaluate functions by substituting the input values into the functions. This means that we can evaluate by substituting into the given function :
Similarly, we can evaluate by substituting into the given function :
We can now evaluate by adding these values together. We get
Before we move on to our next example, there is another useful definition we can consider for functions.
Definition: Root of a Function
We call any input value for which the output of the function is 0 a root of the function.
This means if is a root of the function , then we have . Thus, the roots of a function are all the solutions of the equation . It follows that we can find roots of polynomial functions by factoring the polynomial; the roots of the function are then given by the -values that make one of the factors 0.
Letβs see an example of using factoring to determine the roots of a quadratic function.
Example 2: Finding the Roots of a Quadratic Function by Solving It
Find all of the roots of the function .
Answer
We first recall that the roots of a function are the input values for which the output of the function is 0. This means that to find the roots of , we want to solve the equation . We can substitute in the quadratic we are given to get the equation
This is a quadratic equation, and we recall that we can solve these by factoring. In particular, we want to find two numbers that multiply to give 4 and add to give 5. Since and , we can factor the quadratic as
We now have that the product of two factors is equal to 0. This can only happen if one of the factors is 0. Thus, we either have or . We can solve each equation separately to get or .
Hence, the roots of the function are and .
It is worth noting that we can check that the answers we found are indeed roots by substituting them back into the function. For example, we can verify is a root of by evaluating :
We have verified that , so is a root of .
In our next example, we will determine the roots of a polynomial function by rewriting it as a quadratic function.
Example 3: Finding the Roots of a Disguised Quadratic
Find all the real roots of the function .
Answer
We first recall that the roots of a function are the input values for which the output of the function is 0. This means that to find the roots of , we want to solve the equation . We can substitute in the polynomial we are given to get the equation
There are many ways we could solve this polynomial equation. The easiest way is to note that this is a quadratic equation in since . This allows us to rewrite the equation as
If we now substitute into the equation, we get
This is now a quadratic equation in . We can solve this by factoring. We note that and . This allows us to factor the quadratic as follows:
We now have that the product of two factors is equal to 0. This can only happen if one of the factors is 0. Thus, we either have or . We can solve each equation for to get or .
However, we are not looking for values of ; we need to find values of . So, we substitute into each equation. We get or . Since , we know there are no real solutions for . Thus, we only need to solve . Taking square roots of both sides of the equation yields or .
Hence, the roots of the function are and .
In our next example, we will determine all of the input values for which two given functions are equal.
Example 4: Finding the π₯-Values for Which Two Quadratic Functions Are Equal
If and , find all of the real values of such that .
Answer
To determine the values of such that , we can substitute the given expressions for and into the equation to get
This is a quadratic equation; we can simplify the equation by rearranging and combining like terms. Letβs start by subtracting from both sides of the equation and adding to both sides of the equation:
The right-hand side of the equation simplifies to give 0, and we can combine like terms on the left-hand side of the equation to get
We can now divide the equation through by 2 to get
This is now a quadratic equation in standard form; we can factor this equation by finding two numbers whose product is 3 and whose sum is 4. We note that and . This allows us to factor the quadratic to get
We now have that the product of two factors is equal to 0. This can only happen if one of the factors is 0. Thus, we either have or . We can solve each equation separately to get or .
Hence, the functions are equal if or .
There is one final concept we can consider for quadratic functions. We recall that we can rewrite a quadratic function by completing the square. In general, we can rewrite in the form for some constants and .
We know that squaring a number gives a nonnegative number, so for any real value of , we must have . We can use this to construct an inequality for . We want to multiply the inequality through by , but we do not know the sign of .
If is positive, then
Adding to both sides of the inequality then gives
In other words, for any input value of . We can see that the function achieves this minimum output when .
Similarly, if is negative, then we need to reverse the inequality:
Adding to both sides of the inequality then gives
In other words, for any input value of . We can see that the function achieves this maximum output when .
In our final example, we will find the minimum output of a quadratic function by completing the square.
Example 5: Finding the Minimum Value of a Quadratic Function by Converting It into Vertex Form
By writing the function in the form , find the minimum output of the function.
Answer
To write a quadratic in the form , we need to complete the square. First, we note that the coefficient of is 1; this means we need to find half of the -coefficient. We have . This gives us . If we expand this expression, we get . Therefore, for this expression to be equal to , we need . Thus, and we have
We know that for any input value, . This allows us to see that the smallest output of will occur when . Therefore, the minimum output occurs when and .
Hence, the minimum output of the function is .
In the previous example, we found that the minimum output of a quadratic function of the form occurs when and its value is . In the graph of , this will be the point and it is called the turning point, or vertex, of the quadratic graph.
Letβs finish by recapping some of the important points from this explainer.
Key Points
- A function is a mathematical relationship that takes inputs and maps each input onto exactly one output.
- If is the input, then the corresponding output of function is denoted as .
- The set of all possible input values of a function is called its domain.
- The set of all possible output values of a function is called its range.
- We call any input value for which a root of the function .
- We can find the maximum or minimum output of a quadratic function by completing the square.
