Problem 54 - Precalculus Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(f(x)=\left\\{\begin{array}{ll}6 x-1 & \text { if } x<0 \\ 7 x+3 & \text { if } x \geq 0\end{array}\right.\) a. \(f(-3)\) b. \(f(0)\) c. \(f(4)\)

Step-by-Step Solution

Answer

The evaluations of the piecewise function at the given values are: a. \(f(-3) = -19\), b. \(f(0) = 3\), c. \(f(4) = 31\).

Step 1: Evaluate \(f(-3)\)

For \(x=-3\), since -3 is less than 0, we will use the first function \(6x - 1\). Substitute \(x = -3\) into the first function to get the output. So, \(f(-3) = 6(-3) - 1 = -18 - 1 = -19\).

Step 2: Evaluate \(f(0)\)

For \(x=0\), since 0 is not less than 0, we use the second function \(7x + 3\). Substitute \(x = 0\) into the second function to get the output. So, \(f(0) = 7(0) + 3 = 0 + 3 = 3\).

Step 3: Evaluate \(f(4)\)

For \(x=4\), since 4 is greater than or equal to 0, we will use the second function \(7x + 3\). Substitute \(x = 4\) into the second function to get the output. So, \(f(4) = 7(4) + 3 = 28 + 3 = 31\).

Key Concepts

Evaluating Functions

Understanding how to evaluate functions is a fundamental skill in mathematics. When evaluating functions, you are essentially substiting specific values for the independent variable and calculating the corresponding output. In the case of a piecewise function, like the one in the original exercise, the function is broken into pieces, each defined for a certain interval of the independent variable.

For example, to evaluate the function at the given point \(f(-3)\), you need to determine first which piece of the function applies when \(x = -3\) since multiple expressions might exist for different ranges of \(x\). Once the correct expression is identified, you simply replace \(x\) with -3 and perform the arithmetic operations to find the output. The process is the same for all listed points, but it's crucial to choose the right piece of the function according to the value of \(x\).

Function Notation

Function notation, denoted as \(f(x)\), is a way to represent the outputs of a function with respect to the inputs. In function notation, the letter \(f\) is used to denote the function, and \(x\) represents the independent variable or the input of the function. This notation is concise and makes it easier to refer to functions and their values at specific points.

For instance, \(f(-3)\) indicates the value of the function \(f\) when the independent variable \(x\) is -3. Function notation is not only limited to the use of \(f\) and \(x\); other letters can also be used to name different functions or variables. The main idea is to ensure consistency and clarity when communicating the relationship between variables in mathematical expressions.

Independent Variable

In mathematics, the independent variable is the input or argument of the function, typically represented by the letter \(x\) in function notation. The independent variable is what you can freely change to observe how it affects the value of the function, or the dependent variable, typically represented by \(y\) or \(f(x)\).

In the context of piecewise functions, the independent variable's value determines which piece of the function to use for evaluation. It's 'independent' because its value is not affected by other variables in the function; rather, it's a value that you choose to plug into the function to obtain a result. As seen in the exercise, for different values of the independent variable \(x\), you would use different expressions to evaluate \(f(x)\) demonstrating the integral role the independent variable plays in defining the output of a function.