/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 Find the zero of each function. ... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 58

Find the zero of each function. $$ f(x)=3-4 x $$

Short Answer

Expert verified
The zero of the function is \( x = \frac{3}{4} \).

Step by step solution

01

Identify the function

Given function is: \[ f(x) = 3 - 4x \]
02

Set the function equal to zero

To find the zeros of the function, set \( f(x) = 0 \). So, we have: \[ 3 - 4x = 0 \]
03

Solve for x

Solve the equation for \( x \) to find the zero of the function.\[ 3 - 4x = 0 \]First, subtract 3 from both sides:\[ -4x = -3 \]Then, divide both sides by -4:\[ x = \frac{-3}{-4} \]Simplify to:\[ x = \frac{3}{4} \]

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Key Concepts

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

Linear Functions
A linear function is a type of function that creates a straight line when graphed. It has the general form of \( f(x) = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. In other words, \( m \) tells us how steep the line is, and \( b \) tells us where the line crosses the y-axis.
Linear functions are called 'linear' because they form lines. Their characteristic formula makes them easy to identify and work with. Examples include simple equations like \( f(x) = 2x + 3 \) or \( f(x) = -x + 1 \).
For the given problem, \( f(x) = 3 - 4x \) is a linear function where \( m = -4 \) and \( b = 3 \).
Solving Equations
Solving equations is the process of finding the value of the variable that makes the equation true. With linear equations, this often involves using basic arithmetic operations to isolate the variable on one side of the equation.
To solve an equation like \( 3 - 4x = 0 \), we perform a series of steps:
  • First, move any constants to the other side of the equation.
  • Next, isolate the variable by using inverse operations.

In our example, this involves subtracting 3 from both sides and then dividing by -4:
  • Subtract 3: \( -4x = -3 \)
  • Divide by -4: \( x = \frac{-3}{-4} \)
  • Simplified gives: \( x = \frac{3}{4} \)
Zeros of Functions
The zeros of a function are the values of \( x \) at which \( f(x) = 0 \). In other words, these are the points where the graph of the function crosses the x-axis. Finding the zeros is crucial for understanding the behavior of the function, especially in real-world applications.
For our linear function \( f(x) = 3 - 4x \), setting \( f(x) = 0 \) helps us find the zero of the function.
By solving the equation \( 3 - 4x = 0 \), we find \( x = \frac{3}{4} \). This means the function crosses the x-axis at \( x = \frac{3}{4} \).
Zeros are fundamental in many fields such as engineering, physics, and economics, where they can represent points of equilibrium or critical values.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form so that they are easier to work with. This could mean combining like terms, factoring, or performing arithmetic operations.
In the context of solving equations, simplification can make a problem much more straightforward.
For example, when we solved \( 3 - 4x = 0 \), we simplified by performing operations that isolated \( x \).
  • First, subtracting 3 gave us \( -4x = -3 \).
  • Then, dividing by -4 simplified it further to \( x = \frac{3}{4} \).

Thus, simplification helped us find the solution without extra steps or confusion.

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