Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=15-3 x\)

Step-by-Step Solution

Answer

The real value of \(x\) for which \(f(x)=0\) is \(x = 5\)

Step 1: Write Down the Function

Write down the function \(f(x) = 15 - 3x\)

Step 2: Set the Function Equal to Zero

Set the function equal to zero and solve for \(x\). So the equation is \(0 = 15 - 3x\)

Step 3: Solve for \(x\)

Rearrange the equation to solve for \(x\). In this case, add \(3x\) to both sides of the equation, resulting in \(3x = 15\). Then, divide both sides of the equation by 3, to get \(x = 15 / 3\)

Step 4: Find the Value of \(x\)

Perform the division to find the numerical value of \(x\), thus \(x = 5\)

Key Concepts

Solving Equations

When tackling equations, our main goal is to find the values of the variable that make the equation true. This involves a step-by-step process that requires manipulating the equation while maintaining its balance. For linear equations, which is what we're dealing with in this example, we work with an equation that can be written in the form of \(ax + b = 0\). In this form, \(a\) and \(b\) are constants. Linear equations are distinct for possessing a constant slope and appearing as a straight line when graphed.

To solve the equation, follow these straightforward steps:

• Identify the given equation. In our case, we have \(f(x) = 15 - 3x\).
• Set the equation such that \(f(x) = 0\). This is equivalent to solving \(0 = 15 - 3x\).
• Begin isolation of the variable \(x\) by rearranging terms. Add \(3x\) to both sides to obtain \(3x = 15\).
• Divide both sides by the coefficient of \(x\), which is 3, resulting in \(x = 5\).

Ultimately, the solution \(x = 5\) confirms the value making the equation equal to zero. It's crucial to apply operations such as addition, subtraction, multiplication, or division to both sides equally to solve the equation correctly.

Function Evaluation

A function evaluation involves determining the output of a function for a particular input. Every function defines a relationship between inputs (which can be numbers or variables) and outputs. In mathematical notation, a function is typically represented as \(f(x)\), where \(x\) is the input variable.

For the function \(f(x) = 15 - 3x\), evaluating it means computing the output values for specific \(x\) values. To find where \(f(x) = 0\), we determine which input value causes the output to be zero. This process involves substituting the input value into the function equation and solving for \(x\).

Our task was to identify the input, \(x\), for which the function evaluates to zero, meaning \(f(x)\) becomes exactly zero. Setting \(f(x)\) to zero aids in understanding the relationship between variables and lets us seek meaningful solutions, such as the solution \(x = 5\) derived in this context.

• Input substitutions are key steps in evaluation.
• Understanding function behavior helps in predicting results.

Roots of Equations

Roots of equations refer to the values of the variable that satisfy the equation, effectively making the equation equal to zero. These roots are also called solutions or zeros, as they denote points where a function intersects the x-axis when graphed.

In terms of our specific exercise with the function \(f(x) = 15 - 3x\), finding the root means discovering the value of \(x\) where the function becomes zero, i.e., \(f(x) = 0\). The equation \(15 - 3x = 0\) simplifies to \(x = 5\); hence, \(x = 5\) is the root.

The importance of finding these roots lies in various real-world applications, from calculating the break-even point in business to solving physics problems where the function might represent a physical phenomenon.

• Roots help reveal significant points in functions.
• Knowing roots is essential for graph interpretations.