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Finding the XIntercepts of a Function
(KS4, Year 10)
The Lesson
The xintercepts of a function are found where the graph of a function crosses the xaxis on a pair of Cartesian coordinate axes.How to Find the Xintercepts of a Function
The xintercepts of a function f(x) is found by finding the values of x which make f(x) = 0.Write f(x) = 0, and solve for x to find the xintercepts of a function. The method for solving for x will depend on the type of function (linear, quadratic, or trigonometric etc).
Why Does This Work?
Solving f(x) = 0 uses functional notation (see Note) to express the following idea. The function plotted on the graph is y = f(x). That is, the output of the function at each input x is assigned to y. (The input is the xcoordinate and we write the output as the ycoordinate). The xintercepts are the ouput of the function at the xaxis. Along the xaxis, the value of y is 0. The output f(x) (which is plotted as y) of the function equals 0. The image below shows what we mean:A Real Example of How to Find the Xintercepts of a Function
Finding the xintercepts of a function is easy.Question
What are the xintercepts of the function f(x) = 2x − 4?StepbyStep:
1
Write the function equal to 0, f(x) = 0.
f(x) = 2x − 4 = 0
2
Find the value of x that solves the equation.
What Type of Equation Is It?
This is a linear equation.How Do We Solve It?
Rearrange the equation to find "x = ".2x − 4 = 0  
2x − 4 + 4 = 0 + 4  Add 4 to both sides 
2x = 4  
2x ÷ 2 = 4 ÷ 2  Divide both sides by 2 
x = 2 
Answer:
The xintercept of f(x) = 2x − 4 is 2.Another Real Example of How to Find the Xintercepts of a Function
Question
What are the xintercepts of the function f(x) = x^{2} + 4x + 3?StepbyStep:
1
Write the function equal to 0, f(x) = 0.
f(x) = x^{2} + 4x + 3 = 0
2
Find the value of x that solves the equation.
Read more about how to solve quadratic equations using factoring
What Type of Equation Is It?
This is a quadratic equation.How Do We Solve It?
Factor the quadratic equation into two brackets. Set each bracket to equal 0 and find "x = ".x^{2} + 4x + 3 = 0  
(x + 1)(x + 3) = 0  Find two numbers that multiply to make the constant term (3) and add to the coefficient of x (4) 1 and 3 do this. Put them in brackets added to x 
x + 1 = 0 ⇒ x = −1  Equate 1^{st} bracket to 0 and solve 
x + 3 = 0 ⇒ x = −3  Equate 2^{nd} bracket to 0 and solve 
Read more about how to solve quadratic equations using factoring
Answer:
The xintercepts of f(x) = x^{2} + 4x + 3 are −1 and −3.A Note on Notation
For a function, the input is often denoted by x and the function by f(x), which is equal to the output.When we plot a function on a graph, we make y = f(x). That is, the ycoordinate on the graph is equal to the output of the function.
Different Functions Are SOolved by Different Methods
Functions come in many different forms.
They can be linear:

They can be quadratic:

They can be trigonometic:
How Many Xintercepts?
Different functions have different numbers of xintercepts. Linear functions will have 1 xintercept, unless it is a horizontal line where it will have 0 xintercepts. Quadratic functions can have 0 or 1 or 2 xintercepts. Trigonometric functions can have an infinite number of xintercepts! Do you disagree with something on this page?
 Did you spot a typo?