Problem 50 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

\(f(-1) = -1\), \(f(0) = -2\), \(f(2) = 7\), \(f(4) = 5\).

Step 1: Understand the Function

The function \( f(x) \) is piecewise, meaning it has different expressions based on the value of \( x \). For \( x < 2 \), \( f(x) = x^2 - 2 \). For \( x \geq 2 \), \( f(x) = 4 + |x - 5| \).

Step 2: Evaluate \( f(-1) \)

Since \( x = -1 < 2 \), we use the expression \( f(x) = x^2 - 2 \). Substitute \( x = -1 \): \[f(-1) = (-1)^2 - 2 = 1 - 2 = -1\]

Step 3: Evaluate \( f(0) \)

Since \( x = 0 < 2 \), we use the expression \( f(x) = x^2 - 2 \). Substitute \( x = 0 \): \[f(0) = (0)^2 - 2 = 0 - 2 = -2\]

Step 4: Evaluate \( f(2) \)

Since \( x = 2 \geq 2 \), we use the expression \( f(x) = 4 + |x - 5| \). Substitute \( x = 2 \): \[f(2) = 4 + |2 - 5| = 4 + |-3| = 4 + 3 = 7\]

Step 5: Evaluate \( f(4) \)

Since \( x = 4 \geq 2 \), we use the expression \( f(x) = 4 + |x - 5| \). Substitute \( x = 4 \): \[f(4) = 4 + |4 - 5| = 4 + |-1| = 4 + 1 = 5\]

Key Concepts

Function Evaluation

Function evaluation involves finding the output of a function for specific input values. When dealing with a piecewise function like the given function \(f(x)\), the first step is to determine which part of the function to use for each input value. To evaluate, plug the specific \(x\) value into the appropriate expression in the piecewise definition.

Here's a simple process to follow when evaluating:

• Identify which expression of the piecewise function corresponds to the given input.
• Substitute the input value into this expression.
• Solve the resulting equation to find the output.

Understanding the conditions of the piecewise parts is crucial, as each range specifies when an expression should be used. This is especially important in real-world problems where multiple models might apply under different circumstances.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value function, denoted by \(|x|\), is a key component in evaluating piecewise functions when one of the expressions includes it.

For instance, in the expression \(4 + |x - 5|\), the absolute value \(|x - 5|\) is solved as:

• If \(x - 5\) is positive or zero, \(|x - 5| = x - 5\).
• If \(x - 5\) is negative, \(|x - 5| = -(x - 5)\) or equivalently \(5 - x\).

When evaluating \(f(x)\) at different points, understanding and correctly interpreting the absolute value function is crucial for obtaining the accurate results. It modifies the piecewise expression depending on the value of \(x\), causing the function's graph to exhibit an interesting, often symmetric behavior around the point that causes the expression inside the absolute value to be zero.

Quadratic Functions

A quadratic function is any function that can be written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Quadratic functions typically produce parabolic graphs when plotted. The expression \(x^2 - 2\) in our piecewise function is a simple form of a quadratic function, where the parameter \(a = 1\), \(b = 0\), and \(c = -2\).

In the context of a piecewise function, the quadratic portion dictates the behavior of \(f(x)\) when \(x\) is less than 2. To evaluate, substitute the given \(x\) into \(x^2 - 2\)
as shown in the solution steps. Quadratic functions often appear in many scenarios due to their property of relating to motion under uniform acceleration and describing many natural phenomena. Recognizing a quadratic expression within a more complex function can simplify understanding and solving the problem.