Guide to Finding Zeros of a Function: Explained with an Example

A zero of a function

A zero of a function, also known as a root or solution, is a value of the independent variable (usually denoted as x) for which the function equals zero

A zero of a function, also known as a root or solution, is a value of the independent variable (usually denoted as x) for which the function equals zero. In other words, it is a value that makes the function “vanish” or cross the x-axis on a graph.

To find the zeros of a function, you need to set the function equal to zero and solve for the variable. The resulting solutions will be the zeros of the function. The number of zeros a function may have depends on the degree of the function.

Let’s consider an example to illustrate this concept:

Given the quadratic function f(x) = x^2 – 5x + 6, we want to find the zeros of this function.

Step 1: Set the function equal to zero:
x^2 – 5x + 6 = 0

Step 2: Solve for x. In this case, since we have a quadratic function, we can either factor or use the quadratic formula.

Option 1: Factoring
(x – 3)(x – 2) = 0

Setting each factor equal to zero:
x – 3 = 0 –> x = 3
x – 2 = 0 –> x = 2

Therefore, the function has two zeros: x = 3 and x = 2.

Option 2: Quadratic formula
Using the quadratic formula, which states that for a quadratic equation ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 – 4ac))/(2a)

In our case, a = 1, b = -5, and c = 6. Substituting these values into the quadratic formula, we get:
x = (-(-5) ± √((-5)^2 – 4(1)(6)))/(2(1))
x = (5 ± √(25 – 24))/2
x = (5 ± √1)/2
x = (5 ± 1)/2

This gives us two solutions:
x = (5 + 1)/2 –> x = 6/2 –> x = 3
x = (5 – 1)/2 –> x = 4/2 –> x = 2

Again, we obtain the same zeros: x = 3 and x = 2.

Therefore, in this example, the zeros of the function f(x) = x^2 – 5x + 6 are x = 3 and x = 2.

Remember, if you are given a polynomial function of a higher degree, you may need to use more advanced methods such as factoring, synthetic division, or the rational root theorem to find the zeros.

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