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Chapter 10: Problem 1

What is a zero of a function?

Short Answer

Expert verified
The zero of a function is a value of x for which f(x) equals zero.

Step by step solution

01

- Understand the Concept

A zero of a function is a value of the input (usually denoted as x) that makes the output of the function equal to zero. In other words, it is a solution to the equation f(x) = 0.
02

- Set the Function to Zero

To find the zero of a function f(x), you need to set f(x) equal to zero: f(x) = 0.
03

- Solve for x

Solve the equation f(x) = 0 for x. The solutions will be the zero(s) of the function.
04

- Verify Solutions

Once you have found the potential zeros, plug them back into the original function to verify that they satisfy the equation f(x) = 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function
Before we can understand the zero of a function, we need to know what a function itself is. A function represents a relationship between two sets of elements. By definition, it assigns each input exactly one output. In mathematical terms, if you have a function f defined as f(x), then for every input x in the function's domain, there is a unique output f(x). Functions can be visualized as equations, graphs, or even real-life scenarios, such as converting temperatures from Celsius to Fahrenheit.
Solving Equations
Solving equations is a foundational skill in mathematics that you'll use when finding zeros of functions. An equation states that two expressions are equal. For example, the equation 2x + 3 = 7 tells us that 2x + 3 is equal to 7. Solving an equation means finding the values of the variables that make this statement true. Common techniques for solving equations include isolating the variable, using operations like addition, subtraction, multiplication, and division, and applying algebraic rules such as the distributive property.
Zeros of Functions
Zeros of functions are the specific values of x that make the function's output zero. In other words, if f(x) = 0, then the values of x that satisfy this equation are the zeros of the function. These values are crucial in various mathematical and practical applications, such as identifying roots of polynomials or determining points where a curve intersects the x-axis on a graph. For example, to find the zeros of the function f(x) = x^2 - 4, you set the function to zero: x^2 - 4 = 0. Solving this, you get x = 2 and x = -2.
Verifying Solutions
Once you have potential zeros of a function, how do you confirm they are correct? Verification involves plugging these values back into the original function to see if they indeed make the function's output zero. For instance, if you found x = 2 and x = -2 as zeros of f(x) = x^2 - 4, substitute x=2 into the function: f(2) = 2^2 - 4 = 0 and similarly for x=-2: f(-2) = (-2)^2 - 4 = 0. Since both result in f(x) = 0, these values are verified as correct solutions.

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Most popular questions from this chapter

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+x-2}$$

Find all real roots to each polynomial equation by graphing the corresponding function and locating the x-intercepts. $$x^{4}-12 x^{2}+10=0$$

Find a polynomial equation with real coefficients that has the given roots. $$-1,2,3$$

Find all real zeros to each polynomial function by graphing the function and locating the \(x\) -intercepts. $$ f(x)=x^{3}-0.2 x^{2}-0.05 x+0.006 $$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}+x-1=0$$
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