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Chapter 10: Problem 41

What is the zero of the linear function \(f(x)=m x+b \quad\) with \(\quad m \neq 0 ?\)

Short Answer

Expert verified
The zero of the function is \(x = \frac{-b}{m}\).

Step by step solution

01

Identify the Zero of the Function

To find the zero of the function, we need to determine the value of \(x\) for which \(f(x) = 0\). For the given linear function \(f(x) = mx + b\), this means solving \(mx + b = 0\) for \(x\).
02

Isolate the Linear Term

Set the function equal to zero: \(mx + b = 0\). To isolate the linear term, subtract \(b\) from both sides of the equation to get \(mx = -b\).
03

Solve for x

To solve for \(x\), divide both sides of the equation \(mx = -b\) by \(m\), which yields \(x = \frac{-b}{m}\). This is the value of \(x\) that makes \(f(x) = 0\).

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Key Concepts

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

Zero of a Function
A zero of a function is a value of the variable that makes the function equal to zero. Put simply, if you have a function like this one:
  • For a linear function, the zero is the point where the graph of the function crosses the x-axis.
  • For our linear function \(f(x) = mx + b\), the zero is the \(x\) value that turns the entire function \(f(x) = 0\).
You can think of zeros as the solutions to the equation \(f(x) = 0\). When solving for the zero of a linear function is an important initial point in understanding its behavior, it's almost like finding a starting point.
Solving Equations
Solving equations is a fundamental skill in algebra that involves finding the value of the variable that makes an equation true. Here's a quick overview:
  • First, determine what kind of equation you have. Here, it's a linear equation.
  • Set the equation equal to zero to find the zero of the function.
  • Isolate the variable by performing operations like addition, subtraction, multiplication, or division.
For instance, when solving \(mx + b = 0\):
  • Subtract \(b\) from both sides to start isolating the variable term: \(mx = -b\).
  • Then, divide by \(m\) to solve for \(x\) completely.
These steps help methodically break down the problem for easier understanding.
Linear Equations
Linear equations are equations that, when graphed, produce a straight line. Here's more to know:
  • They follow the form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • The slope \(m\) tells you how steep the line is. The y-intercept \(b\) tells you where the line crosses the y-axis.
Linear equations like \(f(x) = mx + b\) are useful for modeling relationships with constant rates of change. Understanding and solving them are key skills in mathematics, especially since they appear in many real-life scenarios too!

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Most popular questions from this chapter

Differentiate the series $$ \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\cdots $$ and check that the resulting series is the negative of the series for \(\sin x,\) showing (again) that \(\frac{d}{d x} \cos x=-\sin x\)

a. Find the Taylor polynomial at \(x=0\) that approximates \(f(x)=\cos x\) on the interval \(-1 \leq x \leq 1\) with error less than 0.0002 b. Graph \(f(x)=\cos x\) and the Taylor polynomial on \([-2 \pi, 2 \pi]\) by \([-2,2] .\) Use TRACE to estimate the maximum difference between the two curves for \(-1 \leq x \leq 1\).

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Find the Taylor series at \(x=0\) for \(\ln (x+1)\) by integrating both sides of $$ \frac{1}{1+x}=1-x+x^{2}-x^{3}+\cdots \text { for }|x|<1 $$

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