Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(t)=3 t+1$$ (a) \(f(2)\) (b) \(f(-4)\) (c) \(f(t+2)\)

Step-by-Step Solution

Answer

The values of the function are: (a) \(f(2) = 7\), (b) \(f(-4) = -11\), (c) \(f(t+2) = 3t + 7\)

Step 1: Evaluate f(2)

To find \(f(2)\), replace all occurrences of \(t\) in the function with 2: \(f(2) = 3 \cdot 2 + 1 = 6 + 1 = 7\)

Step 2: Evaluate f(-4)

To find \(f(-4)\), replace all occurrences of \(t\) in the function with -4: \(f(-4) = 3 \cdot -4 + 1 = -12 + 1 = -11\)

Step 3: Evaluate f(t+2)

To find \(f(t+2)\), replace all occurrences of \(t\) in the function with \(t+2\): \(f(t+2) = 3 \cdot \left(t+2\right) + 1 = 3t + 6 + 1 = 3t + 7\)

Key Concepts

Independent Variable

In mathematics, an independent variable is a fundamental concept that is used in creating and evaluating functions. The independent variable is the input of the function, which can vary, and is often represented by a symbol such as \(t\), \(x\), or \(y\). In the given exercise, the independent variable is \(t\). The function \(f(t) = 3t + 1\) illustrates how the variable \(t\) affects the value of the function when it is replaced with a specific number or expression.

Understanding the role of the independent variable is crucial because it allows us to explore how changes affect the output of a function. For example, by substituting different values into the function for \(t\) and simplifying, we can see how the function responds and what value it outputs. Evaluating \(f(2)\), \(f(-4)\), and \(f(t+2)\) as shown in the exercise is a practical way of understanding how the independent variable can be manipulated and the resulting changes calculated.

Simplification

Simplification involves performing operations to express a function or an equation in its simplest form. In the context of the given exercise, simplification comes into play after the independent variable is substituted into the function. Once a value or another expression replaces the variable \(t\), the function undergoes simplification to arrive at a straightforward result.

To simplify, follow these steps:

• Substitute the given value for the independent variable.
• Perform the arithmetic operations as indicated in the function.
• Combine like terms to arrive at a simple expression or number.

For example, substituting \(2\) into \(f(t) = 3t + 1\) gives \(f(2) = 3 \times 2 + 1 = 6 + 1 = 7\). This step-by-step process shows how a complex expression can be turned into a simple, understandable result.

Function Notation

Function notation is a way of representing functions in mathematics. It clearly shows the relationship between the input and output. The notation is structured to identify the name of the function and the independent variable. In this exercise, \(f(t) = 3t + 1\) is the function notation.


Function notation uses parentheses to input a specific value or expression in place of the independent variable. For instance, \(f(2)\) indicates that we need to substitute \(2\) for \(t\) in the function. This notation offers a crystal-clear method of writing and evaluating functions, making it easier to convey mathematical ideas efficiently.

• \(f(2)\) implies the independent variable \(t\) is replaced with 2 and evaluated.
• \(f(-4)\) means substituting \(-4\) for \(t\), leading to a different output.
• \(f(t+2)\) signifies evaluating the function as the expression \(t+2\) is substituted for \(t\).

By using function notation, mathematicians can express functions in a way that is both concise and intuitive, which is advantageous for communication and problem-solving.