Problem 15 - Elementary and Intermediate Algebra

Question

Fill in the blank so that the following statements are equivalent: If \(y=5 x+1,\) find the value of \(y\) when \(x=8\). If \(f(x)=5 x+1,\) find \(\square\)

Step-by-Step Solution

Answer

\(f(8) = 41\)

Step 1: Identify the Expression

Given the equation for both statements: first, with dependent variable notation, \(y = 5x + 1\) and second, using a function notation \(f(x) = 5x + 1\). Both represent the same linear relationship between \(x\) and the expression \(5x + 1\).

Step 2: Substitute x in Both Equations

Substitute \(x = 8\) into both forms, \(y = 5(8) + 1\) and \(f(x) = 5(8) + 1\). This shows the input value we need to evaluate the function or expression for.

Step 3: Calculate the Expression

Perform the arithmetic operation by calculating the value \(5 \times 8 + 1 = 40 + 1 = 41\). This gives the output value of \(y\) or \(f(x)\) when \(x = 8\).

Step 4: Fill in the Blank

The equivalent function notation statement is \(f(8) = 41\). Thus, the blank should be filled with \(f(8) = 41\) to make both statements equivalent.

Key Concepts

Dependent Variable

In mathematical equations and functions, the "dependent variable" is the one whose value depends directly on another variable. In the equation \(y = 5x + 1\), the dependent variable is \(y\). It relies on the value of \(x\), meaning that changes in \(x\) will result in changes to \(y\).
In general, dependent variables are a crucial part of understanding how different quantities relate to each other, especially in functions.

• They help us model real-world situations, where the outcome is affected by one or more inputs.
• In function notation, \(f(x)\) represents the output or dependent variable, while \(x\) is usually the independent variable.

Knowing which variable is dependent helps us understand the function's behavior, as it gives insight into how outputs will change with different inputs.

Linear Relationship

The term "linear relationship" describes a relationship between two variables where the change in one variable is proportional to the change in another variable. This is typically represented by a straight line when graphed.
The function \(f(x) = 5x + 1\) is an example of a linear relationship. The equation is in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In our case:

• The slope \(m = 5\) indicates the rate of change. For every unit increase in \(x\), \(y\) increases by 5 units.
• The y-intercept \(b = 1\) denotes where the line crosses the y-axis, meaning when \(x = 0\), \(y = 1\).

Linear relationships are foundational in mathematics, often used because they simplify modeling and predicting behaviors across various disciplines.

Evaluate the Function

"Evaluating the function" means finding the value of the dependent variable for a specific input of the independent variable. For instance, in the function \(f(x) = 5x + 1\), to evaluate, replace \(x\) with a specific number, then compute the resulting expression.
In the given problem, when \(x = 8\), substitute this value into the function:

• First, calculate \(5 \times 8\), which equals 40.
• Then, add 1 to get the final value 41.

Hence, \(f(8) = 41\). Evaluating functions is essential for understanding how outputs change with varying inputs, checking appropriateness of models, and solving real-world problems.