Problem 29 - Algebra and Trigonometry

Question

Evaluate the piecewise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{x^{2}+2 x} & {\text { if } x \leq-1} \\ {x} & {\text { if }-11}\end{array}\right.} \\ {f(-4), f\left(-\frac{3}{2}\right), f(-1), f(0), f(25)}\end{array} $$

Step-by-Step Solution

Answer

The function values are: \( f(-4) = 8 \), \( f\left(-\frac{3}{2}\right) = -\frac{3}{4} \), \( f(-1) = -1 \), \( f(0) = 0 \), \( f(25) = -1 \).

Step 1: Evaluate f(-4)

To find \( f(-4) \), we need to determine which condition \( -4 \) satisfies in the piecewise function. Since \( -4 \leq -1 \), we use the first expression: \( f(x) = x^2 + 2x \). Thus, \( f(-4) = (-4)^2 + 2(-4) = 16 - 8 = 8 \).

Step 2: Evaluate f\left(-\frac{3}{2}\right)

To find \( f\left(-\frac{3}{2}\right) \), we observe that \(-1 < -\frac{3}{2} \leq 1\) is not correct; instead \(-\frac{3}{2} \leq -1\), so we use \( f(x) = x^2 + 2x \). Calculating, we get: \( f\left(-\frac{3}{2} \right) = \left(-\frac{3}{2}\right)^2 + 2\left(-\frac{3}{2}\right) = \frac{9}{4} - 3 = -\frac{3}{4} \).

Step 3: Evaluate f(-1)

For \( f(-1) \), we have \( -1 \leq -1 \), which fits the first condition. Thus, we use \( f(x) = x^2 + 2x \). So, \( f(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1 \).

Step 4: Evaluate f(0)

For \( f(0) \), the condition \( -1 < 0 \leq 1 \) is satisfied, so we use \( f(x) = x \). Therefore, \( f(0) = 0 \).

Step 5: Evaluate f(25)

To find \( f(25) \), we satisfy the third condition since \( 25 > 1 \). Thus, we use \( f(x) = -1 \). Therefore, \( f(25) = -1 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept when dealing with piecewise functions. A piecewise function is composed of multiple sub-functions, each applying to a different part of the function's domain. To evaluate such a function, you begin by identifying which part of the domain your input value falls into. This determines which "piece" or specific rule of the function will be used.

Consider the function given in the exercise:

• When evaluating \( f(-4) \), the value \(-4\) satisfies the condition \( x \leq -1 \). We then use the rule \( f(x) = x^2 + 2x \), resulting in \( f(-4) = 8 \).
• Similarly, for \( f(0) \), since \(-1 < 0 \leq 1\), we apply \( f(x) = x \), which gives us \( f(0) = 0 \).
• For \( f(25) \), \( x > 1 \) is true, so we use \( f(x) = -1 \) leading to \( f(25) = -1 \).

Function evaluation in piecewise functions requires careful checking of conditions to ensure the correct formula is applied for each specific value.

Algebra Concepts

Algebra plays a critical role when evaluating expressions in piecewise functions. Each sub-function might require different algebraic manipulations depending on its formula. For instance, you may need to perform operations like squaring numbers or factoring.

In our exercise, when evaluating \( f(-4) \), we square \(-4\) and add two times that value to solve the expression \( (-4)^2 + 2(-4) = 16 - 8 = 8 \). For \( f\left(-\frac{3}{2}\right) \), we compute \( \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \) and perform subtraction to get \(-\frac{3}{4}\).

These processes require a solid understanding of algebraic techniques such as:

• Squaring Numbers: Multiplying a number by itself.
• Combining Like Terms: Adding or subtracting terms with the same variables.
• Simplifying Fractions: Finding an equivalent fraction using the smallest possible denominators.

Grasping these algebra concepts is essential when working through each section of a piecewise function.

Conditional Functions

Conditional functions provide a unique way to define mathematical functions that can behave differently depending on the input value. These are particularly useful for modeling situations where different rules apply over different intervals.

Each piece of a piecewise function has its own condition, or rule, which tells you when and how to apply it. For example, with our case:

• The first condition, \( x \leq -1 \), uses the expression \( x^2 + 2x \).
• The second condition, \(-1 < x \leq 1\), applies the rule \( f(x) = x \).
• The third condition, \( x > 1 \), uses the constant function \( f(x) = -1 \).

Conditional functional structure gives piecewise functions their character and flexibility. Understanding how to read and interpret these conditions is crucial. For any input, you first determine the interval it belongs to, and then apply the corresponding rule. This ensures that each evaluation is accurate and follows the defined logic of the function.