Problem 45 - College Algebra

Question

Evaluate each piecewise function at the given values of the independent variable. $$\begin{aligned} &f(x)=\left\\{\begin{array}{ll} 3 x+5 & \text { if } x<0 \\ 4 x+7 & \text { if } x \geq 0 \end{array}\right.\\\ &\begin{array}{lll} \text { a. } f(-2) & \text { b. } f(0) & \text { c. } f(3) \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The results for function evaluation are \(f(-2) = -1\), \(f(0) = 7\), and \(f(3) = 19\).

Step 1: Understand the Function

The piecewise function given is \( f(x) = \begin{cases} 3x+5, & \text { if } x < 0 \\ 4x+7, & \text { if } x \geq 0 \end{cases} \) This function tells that if x is less than zero, the function f(x) can be written as \(3x + 5\). If x is greater than or equal to zero, then f(x) can be written as \(4x + 7\).

Step 2: Evaluate f(-2)

For the function \(f(-2)\), as the value of x is less than 0, we'll use the first case of the function. Substitute x = -2 into \(3x + 5\). This leads to: \(-2*3 + 5 = -6 + 5 = -1\).

Step 3: Evaluate f(0)

For the function \(f(0)\), as the value of x is equal to 0, we'll use the second case of the function. Substitute x = 0 into \(4x+7\). This simplifies to \(4*0 + 7 = 0 + 7 = 7\).

Step 4: Evaluate f(3)

For the function \(f(3)\), x is greater than 0, we'll use the second case of the function. Substitute x = 3 into \(4x+7\). This simplifies to \(4*3 + 7 = 12 + 7 = 19\).

Key Concepts

Function Evaluation

Evaluating a function means finding the output value for a given input in the function. In the context of piecewise functions, this process involves determining which piece of the function to use based on the value of the independent variable, which is usually represented as \(x\). Each piece applies to a specific range of \(x\) values. For example, in our piecewise function: \[f(x) = \begin{cases} 3x + 5, & \text{if } x < 0 \ 4x + 7, & \text{if } x \geq 0 \end{cases} \]The function is broken into two expressions:- \(3x + 5\) for \(x < 0\)- \(4x + 7\) for \(x \geq 0\)To evaluate \(f(x)\) at specific points:- Check the value of \(x\) to see which condition it satisfies.- Use the corresponding piece of the function for that \(x\) value.- Perform the arithmetic operation, following the order of operations.

Algebraic Expressions

An algebraic expression contains numbers, variables, and operations. In our piecewise function, each separate assignment \((3x + 5)\) and \((4x + 7)\) is an algebraic expression. These expressions tell us how to compute the function's output:- For \(x < 0\), the expression \(3x + 5\) includes: - A coefficient (3) times the variable (\(x\)). - A constant number (+5).- For \(x \geq 0\), the expression \(4x + 7\) includes: - A coefficient (4) times the variable (\(x\)). - A constant number (+7).When evaluating an algebraic expression:- Substitute the given value of \(x\) into the expression.- Follow the order of operations: multiplication before addition.- Simplify the expression to find the result. For example, to compute \(f(-2)\), we replace \(x\) with \(-2\) in \(3x + 5\), giving us \(-6 + 5 = -1\).

Independent Variable

In mathematics, an independent variable is a variable whose variation does not depend on another variable. It is the input value of a function that we can change. In a piecewise function, the independent variable determines which part of the function to utilize.Let's consider \(x\) as the independent variable in our example. It helps us decide which piece of the piecewise function to apply:- If \(x < 0\), we use the expression \(3x + 5\).- If \(x \geq 0\), we switch to \(4x + 7\).This variability allows functions to model situations with different rules depending on conditions. The independent variable not only helps us select the correct expression, but it is also pivotal in controlling the output value. By manipulating \(x\), we get a sense of how changes affect the overall result. This concept is essential not just for piecewise functions but for all functions, as \(x\) guides us in evaluating and understanding the behavior of mathematical relationships.