Problem 41 - Brief Applied Calculus

Question

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\left\\{\begin{array}{ll} x-2 & \text { if } x<3 \\ 7-4 x & \text { if } x \geq 3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is a piecewise linear function.

Step 1: Recognize the Structure

The function provided is in a piecewise format, meaning it is defined by different expressions depending on the value of \( x \). This suggests that it may be a piecewise linear function, but we need to examine each piece to rule out other possibilities like polynomial, rational, or exponential functions.

Step 2: Analyze Each Piece

Let's look at the first piece of the function, \( f(x) = x - 2 \) for \( x < 3 \). This is a linear expression, as it is of the form \( ax + b \) with \( a = 1 \) and \( b = -2 \). Now, examine the second piece, \( f(x) = 7 - 4x \) for \( x \geq 3 \), which is also a linear expression, as it fits the form \( ax + b \) with \( a = -4 \) and \( b = 7 \).

Step 3: Determine the Overall Function Type

Since both individual expressions that make up the function are linear, and they are defined over different intervals, the entire function is a combination of linear expressions on different domains. This makes the function a piecewise linear function.

Key Concepts

Understanding Function Types

Functions come in various forms, each with unique characteristics. Identifying the type of function helps us understand its behavior.

- **Polynomial Functions:** These functions are made from sums of power functions of the form \( ax^n \), where \( a \) is a coefficient, and \( n \) is a non-negative integer. They have smooth, continuous graphs without any breaks or jumps.- **Rational Functions:** They are ratios of two polynomials. The form is \( \frac{P(x)}{Q(x)} \), where neither \( P(x) \) nor \( Q(x) \) is zero. These functions can have breaks or asymptotes, unlike polynomials.- **Exponential Functions:** These involve constants raised to variable powers, such as \( a^x \), where \( a \) is a positive constant. They grow rapidly and their graphs show a consistent exponential curve.

In contrast, **piecewise linear functions** like \( f(x) = \begin{cases} x - 2, & \text{if } x < 3 \ 7 - 4x, & \text{if } x \geq 3 \end{cases} \), are formed from linear expressions defined over different intervals. They often have distinct segments that line up on the coordinate plane, sometimes resulting in a graph with sharp changes in direction.

Exploring Calculus

Calculus primarily deals with change and motion, focusing on two main ideas: differentiation and integration. When dealing with piecewise linear functions, calculus can reveal crucial characteristics like slopes and points of discontinuity.

- **Differentiation** involves finding the derivative, which indicates the rate of change of a function. For each piece in a piecewise linear function, the derivative is just the slope of the linear segment. - **Integration** determines the area under the curve of a function. With piecewise functions, you integrate each segment independently, accounting for changes in expressions.

In calculus, understanding how piecewise functions behave at their boundaries is essential. This helps in calculating limits and understanding any discontinuity that might arise at the points where the function's rule changes.

Breaking Down Mathematical Expressions

Mathematical expressions represent quantities symbolically, using numbers, operators, and variables. Breaking them down into simpler pieces helps in understanding and solving complex equations.

Piecewise functions simplify expression complexity by dividing a function into segments. For example, \( f(x) = x - 2 \) when \( x < 3 \) and \( f(x) = 7 - 4x \) when \( x \geq 3 \) exemplifies how expressions can vary depending on conditions.

- **Linear Expressions** are straightforward, of the form \( ax + b \), where \( a \) is the slope and \( b \) is the y-intercept.- **Evaluating Expressions** within a piecewise function requires identifying which segment applies for a specific value of \( x \), ensuring correct computation.

For clarity, graphing these expressions shows how different linear expressions interact in visual forms, aiding in deeper comprehension of how the overall function behaves through its segments.