Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=2 x-3\) (a) \(f(1)\) (b) \(f(-3)\) (c) \(f(x-1)\) (d) \(f\left(\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

Using the function \(f(x)=2x-3\), the values are \(f(1)=-1\), \(f(-3)=-9\), \(f(x-1)=2x-5\), \(f(1/4)=-2.5\)

Step 1: Evaluate the function f at x = 1

Substitute \(x = 1\) into the given function \(f(x) = 2x-3\). Calculate \(f(1) = 2(1) - 3 = -1\)

Step 2: Evaluate the function f at x = -3

Substitute \(x = -3\) into the function. Calculate \(f(-3) = 2(-3) - 3 = -9\)

Step 3: Evaluate the function f at x = x - 1

Substitute \(x = x - 1\) into the function. This results in \(f(x - 1) = 2(x - 1) - 3 = 2x - 2 - 3 = 2x - 5\)

Step 4: Evaluate the function f at x = 1/4

Substitute \(x = 1/4\) into the function. Calculate \(f(1/4) = 2 (1/4) - 3 = -2.5\)

Key Concepts

Independent Variable

In mathematics, when working with functions, understanding the concept of an independent variable is crucial. The independent variable is the input of the function, which you change to see how the output, or dependent variable, reacts.
For the function given in the exercise, which is \( f(x) = 2x - 3 \), the independent variable is \( x \). You can think of \( x \) as a placeholder that can take any value from its domain.
It's where you "plug in" values, like 1, -3, or \( \frac{1}{4} \), to see what the function does with them. This helps determine how the function behaves for different values, giving insights into its characteristics and graph.

Function Simplification

Function simplification often involves taking a complex expression and making it easier to work with. This process is important for understanding the essence of what the function does and for making further calculations easier.
In the given example, each time a value is substituted into the function \( f(x) = 2x - 3 \), you perform a simplification. Consider how each of the specific values, like \( x = 1 \), \( x = -3 \), and so forth, were plugged into the function:

• Substitute \( x = 1 \)
• Simplify by calculating \( f(1) = 2(1) - 3 = -1 \)
• Substitute \( x = -3 \)
• Simplify by calculating \( f(-3) = 2(-3) - 3 = -9 \)
• Use another substitution \( x = x - 1 \)
• Simplify to get \( f(x - 1) = 2(x - 1) - 3 = 2x - 5 \)

These steps help you to directly see the impact of changes in the independent variable on the function’s output.

Function Substitution

Function substitution is a fundamental process in evaluating functions, where you replace the independent variable with a specific value or expression. This technique allows you to find out what the function outputs for that specific input.
In the exercise example, you perform substitutions such as substituting \( x = 1 \) or \( x = -3 \) into the function \( f(x) = 2x - 3 \). Similarly, you can substitute an expression like \( x = x - 1 \) to discover how the function operates under a shifted input.
The steps are straightforward:

• Choose the value or expression to substitute in place of \( x \).
• Replace \( x \) in the function with your chosen input.
• Calculate the resulting expression or number.

Through substitution, you reveal how different inputs transform to outputs, providing insights into the changes in function behavior.