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Chapter 4: Problem 29

Determine the zeros, if any, of each function. $$ f(x)=3 x+7 $$

Short Answer

Expert verified
The zero of the function is \( x = -\frac{7}{3} \).

Step by step solution

01

- Set the Function Equal to Zero

To find the zeros of the function, set the function equal to zero: \[ f(x) = 3x + 7 = 0 \]
02

- Solve for x

Isolate the variable x. Start by subtracting 7 from both sides: \[ 3x + 7 - 7 = 0 - 7 \]This simplifies to: \[ 3x = -7 \]
03

- Divide Both Sides by 3

Finally, divide both sides of the equation by 3 to solve for x: \[ x = \frac{-7}{3} \]

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Key Concepts

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

linear equations
Linear equations are mathematical expressions that form straight lines when plotted on a graph. They are written in the form: \(ax + b = 0\), where \(a\) and \(b\) are constants.
A function like \(f(x) = 3x + 7\) is linear because it represents a straight line on a graph. Understanding linear equations is essential because they are foundational in algebra.
Here are a few key points about linear equations:
  • They graph as straight lines.
  • The coefficient of \(x\) (in this case, 3) determines the slope of the line.
  • The constant term (in this case, 7) determines where the line crosses the y-axis.
Solving linear equations involves finding the value of \(x\) that makes the equation true. This is important in various fields such as physics, engineering, and economics.
solving for variables
Solving for variables means finding the value of the unknown that satisfies the equation.
In the context of the original exercise, the variable we are solving for is \(x\) in the function \(f(x) = 3x + 7\).
Here's a step-by-step breakdown:
  • First, understand the equation. What are you solving for?
  • Rearrange the equation to isolate the variable on one side.
  • Perform arithmetic operations like addition, subtraction, multiplication, or division to solve for the variable.
In our example, subtract 7 from both sides to get \(3x = -7\). Then, divide both sides by 3 to isolate \(x\). This gives us \(x = \frac{-7}{3}\).
set the function equal to zero
Setting the function equal to zero is a technique used to find the zeros of the function. The zeros are the values of \(x\) where the function crosses the x-axis.
In other words, they are the points where the output of the function is zero. For our function \(f(x) = 3x + 7\), we set: \(3x + 7 = 0\).
This step is essential because it allows us to transform the function into a more solvable form: an equation we can work with.
Setting the function to zero simplifies the process of finding where the function intersects the x-axis.
isolating the variable
Isolating the variable means getting the variable by itself on one side of the equation. This helps in determining the variable's value. In our example function \(f(x) = 3x + 7\), isolating the variable involves a few steps:
  • First, subtract 7 from both sides to remove the constant term from the left side: \(3x + 7 - 7 = 0 - 7\).
  • This simplifies to: \(3x = -7\).
  • Next, divide both sides by the coefficient of \(x\), which is 3: \(3x \div 3 = -7 \div 3\), resulting in: \(x = \frac{-7}{3}\).
Isolating the variable helps simplify the equation step-by-step, making it easier to understand and solve.

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