Problem 38 - College Algebra

Question

Evaluate each piecewise 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 function evaluates to -19 when \(x = -3\), 3 when \(x = 0\), and 31 when \(x = 4\).

Step 1: Evaluate the function at \(x = -3\)

As given, \(x\) is less than zero in this step, so use the formula for that condition, which is \(6x - 1\). Substituting, you get: \(f(-3) = 6 * -3 - 1 = -18 - 1 = -19\)

Step 2: Evaluate the function at \(x = 0\)

In this step, \(x\) is zero (0), so use the formula for \(x\) greater than or equal to zero, which is \(7x + 3\). Substituting, you get: \(f(0) = 7 * 0 + 3 = 0 + 3 = 3\)

Step 3: Evaluate the function at \(x = 4\)

Again here, \(x\) is greater than zero, so use the formula for \(x\) greater than or equal to zero (\(7x + 3\)). Substituting, you get: \(f(4) = 7 * 4 + 3 = 28 + 3 = 31\)

Key Concepts

Evaluating Functions

Understanding how to evaluate functions is a foundational skill in algebra that allows students to determine the output of a function for a given input. Specifically, when dealing with piecewise functions, this requires identifying which part of the function to use based on the value of the independent variable, in this case, x.

In the provided textbook exercise, the function f(x) changes its formula depending on whether x is less than zero or greater than or equal to zero. To evaluate f(x) for a specific value, you must first determine which conditional expression applies and then substitute the value into the correct algebraic function.

• For f(-3), since -3 is less than zero, we use the formula 6x - 1.
• For f(0), as 0 is neither less than zero nor greater, but exactly zero, we follow the convention for x ≥ 0 and use 7x + 3.
• Similarly, for f(4), since 4 is greater than zero, the formula 7x + 3 is appropriate here as well.

Ensuring clarity in each of these steps and providing simplistic examples can greatly help students grasp the concept of evaluating functions.

Conditional Expressions

Conditional expressions are the 'if-then' statements in mathematics that define the rules for which parts of a piecewise function to use when evaluating it. In our context, the conditionals are x < 0 and x ≥ 0, helping to segment the function into parts that apply to different intervals of x.

When a piecewise function is provided, it essentially means that the function has different definitions depending on the input values. Understanding these conditions is crucial for correctly evaluating the function, as seen with f(x).

• If the input value meets the condition x < 0, then the first expression 6x - 1 is used.
• If it meets the condition x ≥ 0, then the second expression 7x + 3 takes precedence.

It's essential to identify the right conditional to apply to the input value before proceeding with the evaluation.

Algebraic Functions

Algebraic functions consist of operations that include addition, subtraction, multiplication, division, powers, and roots applied to variables. They form the backbone of evaluating functions. In the provided exercise, we have two algebraic expressions: 6x - 1 and 7x + 3. Both are linear equations, which are the simplest type of algebraic functions.

Evaluating algebraic functions involves substituting the variable with a number and performing the arithmetic operations. For example, when x = -3, we substitute -3 into 6x - 1 to calculate f(-3). By honing in on these algebraic manipulation skills, students can solve a wide range of problems within algebra, including those involving piecewise functions.

• In 6x - 1, the operation is straightforward – multiply x by 6 and subtract 1.
• In 7x + 3, similarly, multiply x by 7 and add 3 for the final result.

These functions demonstrate how basic algebra serves as a vital tool for evaluating more complex, piecewise functions.