Problem 25 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 34, the function is the set of all ordered pairs \((x, y)\) satisfying the given equation. Find the domain and range of the function, and draw a sketch of the graph of the function. $$ f: y=|x|+|x-1| $$

Step-by-Step Solution

Answer

Domain: \( (-\infty, \infty) \). Range: \( [1, \infty) \).

Step 1 Title - Identify the given function

The given function is \( y = |x| + |x-1| \). This function consists of the sum of two absolute value expressions.

Step 2 Title - Determine the domain

The domain of a function is the set of all possible input values (x-values). Since there are no restrictions on the values that x can take in the function \( y = |x| + |x-1| \), the domain is all real numbers, \( \text{Domain} = (-\infty, \infty) \).

Step 3 Title - Analyze the function to find critical points

The function \( y = |x| + |x-1| \) has critical points where the expressions inside the absolute values change signs: at \( x = 0 \) and \( x = 1 \). These points need to be analyzed separately.

Step 4 Title - Solve the function in different intervals

Divide the function into three intervals to solve each piece: 1. \(x < 0\) : \( y = -x + 1 - x = -2x + 1 \) 2. \(0 \le x < 1\) : \( y = x + 1 - x = 1 \) 3. \(x \ge 1\) : \( y = x + x - 1 = 2x - 1 \)

Step 5 Title - Determine the range

The range of the function is the set of all possible output values (y-values). By analyzing the function over the determined intervals, the lowest y-value is 1 (occurring in the interval \(0 \le x < 1\)). As x increases or decreases away from the interval, y-values grow without bound. Thus, the range is \( [1, \infty) \).

Step 6 Title - Draw the graph

Plot the function over the three intervals determined in Step 4. Connect the points smoothly, ensuring to highlight changes at \( x = 0 \) and \( x = 1 \). The graph consists of two linear parts meeting at these critical points.

Key Concepts

Introduction to Absolute Value Functions

Absolute value functions are important concepts in calculus and algebra.
They can be recognized by the absolute value symbol: \(|...|\). An absolute value of a number is its distance from zero on the number line, no matter the direction.
This means \(|-3| = 3\) and \(|3| = 3\).
When dealing with functions, absolute values create V-shaped graphs that can split into different linear expressions depending on the sign of the input value.
For example, let's consider the function \(f(x) = |x|\).

• If \(x \text{ is positive}\), \(f(x) \text{ will simply be x}\).
• If \(x \text{ is negative}\), \(f(x) \text{ will be -x}\).

In our original problem, we have two absolute value terms \(y = |x| + |x-1|\).
This adds complexity by creating different behaviors in various intervals of x.

Understanding Domain and Range

In mathematics, the domain and range of a function are fundamental concepts.
The domain refers to the set of possible input values (x-values) for which the function is defined.
The range indicates the set of possible output values (y-values) the function can produce.

• Domain: For most functions, especially polynomials and absolute value functions, the domain is all real numbers, represented as \((-\infty, \infty)\).
• For \(f: y=|x| + |x-1|\) as given, x can take any value without restriction.

Determining the range requires analyzing the function's behavior across its domain.
For the function \(y = |x| + |x-1|\), the lowest y-value is 1.
This is observed within the interval \(0 \leq x < 1\).
As x moves away from this interval either in the positive or negative direction, y increases without limit. Thus, the range of this specific function is given by \([1, \infty)\).

Graphing Piecewise Functions

A piecewise function is defined by multiple sub-functions, each applying to a certain interval of the domain.
With \(y=|x| + |x-1|\), critical points split the function into different linear expressions.
This makes the graph consist of different line segments.
Here's how we can work through it:

• For \(x < 0\), the function simplifies to \(y = -2x + 1\).
• For \(0 < \leq x < 1\), the function simplifies to a constant, \(y = 1\).
• For \(x \geq 1\), the function is \(y = 2x - 1\).

To plot these:

• Start at \(x = 0\), the function is 1. Plot this point.
• For points left of 0, use the expression for \(x < 0\), which gives a linear decrease.
• For points right of 1, use the expression for \(x \geq 1\), which linearly increases.

Connecting these points, making sure to change direction at critical points, gives you the complete graph.
This way, graphing piecewise functions can become straightforward with step-by-step simplifications and interval analysis.