Finding the zero of a function is one of the most frequently encountered problems in basic and advanced algebra classes. These entities are the values of x where f(x) is equal to zero. As you progress and improve your solving skills for the zeroes of functions, you will encounter problems of varying complexity.
You regularly find zeros of functions not only in your math classes but in daily life as well. For example, to find out the amount you must sell to break even, you inevitably need to find the zeroes of the equation you’ve put up. In addition, there are many other models and problems which mandate you to find f(x) zeroes.
Considering the widespread application of functions and their zeroes, you need to master the art of manipulating various equations and expressions to find their zeroes. In this guide, I will help you with:
- Knowing what a function’s zero represents.
- Finding the zeroes of commonly used functions.
- Using a function’s graph to identify its zeroes.
We shall now begin by understanding the basic definition of a zero.
What is the zero of a function?
By understanding what zeroes of a function represent, you can know when to find the zeroes of functions and learn how to find them with the help of a function’s graph. As you go through problems, you will notice that the zeroes of a function could potentially come in several forms. However, there is one important condition – as long as it returns a y-value of 0, you can count it as the function’s zero.
The zeroes of a function are generally defined as the value of x when f(x) is equal to zero. In other words, when f(x) = 0, x is called a zero of the function. Thus, it’s quite easy to see how they got their name. Similarly, when the graph passes through x = a, we can say that a is a zero of the function. Thus, (a,0) is a zero of a function. Given below are a few more examples:
- The function g(x) = x2 – 2 has two different zeros – x = -2 and x = 2, implying that f(-2) = 0 and f(2) = 0.
- The function f(x) = x + 1 has a zero at x = -1 because f(-1) = 0.
- The graph if h(x) passes through (-3,0); thus, x = -3 is a zero of h(x) and h(-3) = 0.
In the graph of a function, its real zeros are represented by the x-intercepts. This is explained by the fact that zeroes are the values of x when y or f(x) is equal to zero. However, there are some cases where the graph does not pass through the x-intercept. This does not mean that the function has no zeroes; rather, it implies that the functions’ zeroes could be complex in nature.
How to find the zeroes of a function?
Depending on the kind of expression you’re dealing with, finding the zeroes of a function can vary in terms of difficulty. It can be as simple as isolating x to one side of the equation and as complicated as repeatedly manipulating the expression to achieve the required result.
Usually, we can find the zeroes of a function f(x) by setting the function to 0. In this case, the zeroes of the function are the values of x that represent the set equation. Thus, to find the zeroes, you must find the values of x where f(x) = 0.
How to find the zeroes of a quadratic function?
There are several complex equations that you can eventually reduce to quadratic equations. For this reason, intermediate algebra classes generally dedicate a lot of time to teaching students about the zeroes of quadratic functions.
You can find the zeroes of a quadratic function by equating the given equation to zero and solving for the values of x that satisfy the equation. While doing so, there are certain important points you need to bear in mind:
- Remember that a quadratic function can have a maximum of two zeroes.
- Before solving, ensure that the quadratic equation is in the standard form (ax2 + bx + c = 0).
- You may factor whenever possible, but go ahead and use the quadratic formula if needed too.
Here is a table of certain valuable strategies for finding the zeroes of quadratic functions, along with indications for their use:
|Is the quadratic equation factorable?||Solve the equation using factoring techniques.|
|Is the quadratic equation not factorable?||Use the quadratic formula to solve the equation.|
|Does the given equation exhibit special algebraic properties?||Use the perfect square trinomial or difference of two square to solve the equation.|
How to find the zeroes of a polynomial function?
You can use the above process for polynomial functions as well; simply equate the polynomial function to zero and find out the values of x that satisfy the given equation. Here is a helpful flowchart for helping you find the best strategy in this regard.
How to find the zeroes of a rational function?
Rational functions are defined as functions that possess a polynomial expression on both their numerator and denominator. You can equation a rational function to zero by applying the same principle when finding other functions’ zeroes.
How to find the zeroes of other functions?
I feel it’s readily apparent now that the same rule applies to all types of functions. When dealing with a special function, you must remember to equate its expression to 0 to find its zeroes.
Given below are some more functions that you might have come across before:
|Type of function||Example|
|Logarithmic function||f(x) = log22x|
|Power function||f(x) = 3x1/3|
|Exponential function||f(x) = 2x+1|
|Trigonometric function||f(x) = -3 sin x|
The zeroes from any of the functions mentioned above will return the values of x where the function is 0. When you are studying the graphs of these functions, you can find their real zeroes by inspecting the graph’s x-intercepts.
I have tried to help you understand the zeroes of a function and how you can find the zeroes of commonly encountered functions. After reading this, make sure you strengthen your newly acquired knowledge by practicing problems related to the same.