Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=\left\\{\begin{array}{ll}3 x-1, & x<0 \\ 2 x+3, & x \geq 0\end{array}\right.\) (a) \(f(-1)\) (b) \(f(0)\) (c) \(f(-2)\) (d) \(f(2)\)

Step-by-Step Solution

Answer

Following the above steps, (a) \(f(-1) = -4\), (b) \(f(0) = 3\), (c) \(f(-2) = -7\), and (d) \(f(2) = 7\).

Step 1: Evaluate f(-1)

For the given value -1, which is less than 0, the first equation \(\3x - 1\) applies. Substitute -1 for x, to get \(3*(-1) - 1 = -3 - 1 = -4\).

Step 2: Evaluate f(0)

For the value 0, which is equal to 0, the second equation \(2x + 3\) applies. Substitute 0 for x, to get \(2*0 + 3 = 0 + 3 = 3\).

Step 3: Evaluate f(-2)

For the value -2, which is less than 0, the first equation \(3x - 1\) applies again. Substitute -2 for x, to get \(3*(-2) - 1 = -6 - 1 = -7\).

Step 4: Evaluate f(2)

Finally, for the value 2, which is greater than 0, the second equation \(2x + 3\) applies. Substitute 2 for x, to get \(2*2 + 3 = 4 + 3 = 7\).

Key Concepts

Independent Variable

When dealing with functions, it's crucial to understand what an independent variable is. In a function, the independent variable is the input, often represented by \(x\), that the function depends upon.
It can be seen as the "cause" that influences the "effect," which is the function's output or the dependent variable.For example, in the piecewise function \(f(x)=\left\{\begin{array}{ll}3x-1, & x<0 \ 2x+3, & x \geq 0\end{array}\right.\), \(x\) is the independent variable.
The value of \(x\) determines which equation within the piecewise function to use.

• If \(x\) is less than 0, use the equation \(3x - 1\).
• If \(x\) is greater than or equal to 0, use the equation \(2x + 3\).

Understanding which part of the function to use relies entirely on recognizing the specific value of the independent variable.

Function Evaluation

Function evaluation is the process of finding the output of a function given a specific input value for its independent variable. In our example, we need to evaluate the function \(f(x)\) at different values of \(x\): -1, 0, -2, and 2.
To do this, follow these straightforward steps:- Identify which equation to use based on the input value. - Substitute the input value into the appropriate equation.- Perform the mathematical operations to find the output.For instance, for \(f(-1)\): Since -1 is less than 0, use the equation \(3x - 1\). Substitute -1, calculate \(3(-1) - 1\), and get \(-4\)
For \(f(0)\): With 0 being equal to 0, utilize \(2x + 3\). Substitute 0, compute \(2*0 + 3\), resulting in 3.By systematically following these steps, evaluating a piecewise function becomes a straightforward task.

Simplification

Simplification is the process of reducing complex expressions into simpler forms. In the context of our piecewise function, once substitutions are made, simplification helps to arrive at the final answers.
When you substitute values into the function, you perform operations like multiplication, addition, or subtraction to get a simpler numeric outcome. Here's a quick rundown:

• For \(f(-1)\): Step from \(3(-1) - 1\) to \(-3 - 1\), simplifies directly into \(-4\).
• For \(f(0)\): Move from \(2(0) + 3\) to \(0 + 3\), simplifies into \(3\).
• For \(f(-2)\): Go from \(3(-2) - 1\) to \(-6 - 1\), simplifies into \(-7\).
• For \(f(2)\): Transition from \(2(2) + 3\) to \(4 + 3\), simplifying produces \(7\).

Simplification removes excess complexity, allowing you to focus on the numerical result. It ensures that the function's output is clear and directly tied to using the correct operations on the independent variable's value.