Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, given each function \(f,\) evaluate \(f(-1), \quad f(0), \quad f(2),\) and \(f(4) .\) \(f(x)=\left\\{\begin{array}{ccc}5 x & \text { if } & x<0 \\ 3 & \text { if } & 0 \leq x \leq 3 \\ x^{2} & \text { if } & x>3\end{array}\right.\)

Step-by-Step Solution

Answer

\(f(-1) = -5\), \(f(0) = 3\), \(f(2) = 3\), \(f(4) = 16\).

Step 1: Identify the Appropriate Case

For each value we need to evaluate, determine which condition of the piecewise function the value falls under. This allows us to know which function rule to apply.

Step 2: Evaluate at f(-1)

Since \(-1 < 0\), use the first piece, \(f(x) = 5x\). Substitute \(-1\) for \(x\):\[f(-1) = 5(-1) = -5\]

Step 3: Evaluate at f(0)

Since \(0\) falls within the interval \([0, 3]\), use the second piece, \(f(x) = 3\). Therefore, \(f(0) = 3\).

Step 4: Evaluate at f(2)

Since \(2\) falls within the interval \([0, 3]\), use the second piece, \(f(x) = 3\). Therefore, \(f(2) = 3\).

Step 5: Evaluate at f(4)

Since \(4 > 3\), use the third piece, \(f(x) = x^2\). Substitute \(4\) for \(x\):\[f(4) = 4^2 = 16\]

Step 6: Compile the Results

We evaluated all necessary values, resulting in \(f(-1) = -5\), \(f(0) = 3\), \(f(2) = 3\), and \(f(4) = 16\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics where you determine the output of a function. To evaluate a function, you need to substitute a given input value (usually represented by the variable \(x\)) into the function. This process helps us understand the behavior and pattern of the function at different points.
For example, if you have a function \(f(x) = 2x + 3\) and you want to evaluate it at \(x = 5\), you simply replace \(x\) with \(5\) in the equation and compute.

• Substitute \(x = 5\) into the function: \(f(5) = 2(5) + 3 = 10 + 3 = 13\).

Through function evaluation, you can analyze how the function responds to changes in the input values.

Piecewise-Defined Functions

Piecewise-defined functions are special types of functions composed of multiple sub-functions. Each sub-function applies to a specific interval of the main function's domain. Essentially, a piecewise function is like having different rules for different sections of the input numbers.
For example, consider the piecewise function used in the exercise:

• \(f(x) = 5x\) if \(x < 0\)
• \(f(x) = 3\) if \(0 \leq x \leq 3\)
• \(f(x) = x^2\) if \(x > 3\)

Each of these parts works for certain ranges of \(x\), indicating what computation should occur based on the input value. When working with piecewise-defined functions, it is crucial to first determine which portion of the function should be used for a given input by examining the conditions provided.

Evaluating Functions at Specific Points

Evaluating functions at specific points involves substituting the point into the appropriate piece of a piecewise-defined function. This requires both identifying which interval the point belongs to and using that sub-function to compute the result.
For the piecewise function \(f(x)\) from the step-by-step solution, evaluating at specific points means:

• For \(f(-1)\), since \(-1\) is less than \(0\), use the first piece \(5x\), resulting in \(f(-1) = -5\).
• For \(f(0)\), since \(0\) falls in the interval \([0, 3]\), use \(f(x) = 3\), yielding \(f(0) = 3\).
• For \(f(2)\), which also lies between \([0, 3]\), again use \(f(x) = 3\), therefore \(f(2) = 3\).
• For \(f(4)\), with \(4 > 3\), apply the third piece \(x^2\), giving \(f(4) = 16\).

This example shows how evaluating functions at specific points allows us to navigate different rules within a piecewise function, ensuring we compute accurate outputs based on given inputs.