Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(w)=\sqrt{w-1}\)

Step-by-Step Solution

Answer

The function is neither even nor odd. Sketch starts at (1,0) and opens rightward.

Step 1: Determine Evenness

A function is even if it satisfies the condition \(f(-w) = f(w)\) for all \(w\) in its domain. Calculate \(f(-w)\):\[ f(-w) = \sqrt{-w-1} \]Since \(-w - 1\) is not equal to \(w - 1\), \(f(-w) eq f(w)\). Hence, the function is not even.

Step 2: Determine Oddness

A function is odd if it satisfies the condition \(f(-w) = -f(w)\) for all \(w\) in its domain. Calculate \(-f(w)\):\[ -f(w) = -\sqrt{w-1} \]The condition for oddness requires \(\sqrt{-w-1} = -\sqrt{w-1}\). Since squaring both sides gives different expressions, \( f(-w) eq -f(w)\), hence the function is not odd.

Step 3: Consider Function's Domain

The function \(f(w) = \sqrt{w - 1}\) is only defined for \(w \geq 1\) because the square root of a negative number is not real. This limits the range of values we can check for symmetry further, confirming that it is neither even nor odd.

Step 4: Sketch the Graph

The graph of \(f(w) = \sqrt{w-1}\) starts from the point \((1,0)\) and increases steadily as \(w\) increases, forming a rightward-opening curve. This curve lies entirely in the right half of the coordinate plane.

Key Concepts

even function

An **even function** is characterized by its symmetry about the y-axis. This simply means that if you were to fold the graph along the y-axis, both halves would match perfectly. Mathematically, a function \( f(x) \) is even if for every x within its domain, the equation \( f(-x) = f(x) \) holds true. This implies that both the negative and positive x values result in the same output when plugged into the function.

For example, consider the classic case of the quadratic function \( f(x) = x^2 \), which is an even function because \( f(-x) = (-x)^2 = x^2 \).

• Symmetry: About the y-axis
• Condition: \( f(-x) = f(x) \)
• Examples: \( x^2, \cos(x) \)

For the function given in the exercise, \( f(w) = \sqrt{w-1} \), this condition does not hold, as \( f(-w) \) results in \( \sqrt{-w-1} \), which is not equal to \( f(w) \). Therefore, this function is not even.

odd function

An **odd function** has a different type of symmetry, which is around the origin. This means that rotating the graph 180 degrees about the origin will yield the same graph. Mathematically, a function \( f(x) \) is odd if for every x within its domain, the relation \( f(-x) = -f(x) \) is satisfied.

A classic example is the cubic function \( f(x) = x^3 \), because \( f(-x) = (-x)^3 = -x^3 = -f(x) \).

• Symmetry: Rotational about the origin
• Condition: \( f(-x) = -f(x) \)
• Examples: \( x^3, \sin(x) \)

In the case of the function \( f(w) = \sqrt{w-1} \), the calculation in the original solution shows that \( \sqrt{-w-1} eq -\sqrt{w-1} \). Hence, the function is not odd.

function domain

The **domain of a function** describes all the possible input values (or x-values) for which the function is defined and gives a real number output. Determining the domain is crucial as it outlines the limit within which we can explore the function's behavior.

For the function \( f(w)=\sqrt{w-1} \), the domain includes all values of \( w \) for which the expression under the square root is non-negative. This means \( w-1 \geq 0 \), simplifying to \( w \geq 1 \). Thus, the domain is all real numbers greater than or equal to 1.

• Condition: \( w-1 \geq 0 \)
• Domain: \( w \geq 1 \)

Understanding the domain helps in determining where the function is graphically represented, and it sets the boundary for analyzing the function's evenness or oddness.

graph sketching

**Graph sketching** involves plotting the function on a set of axes, visually representing the relationship between inputs and outputs. Accurately sketching a graph requires understanding components like intercepts, asymptotes, and critical points to give a more complete graphical picture.

For the function \( f(w) = \sqrt{w-1} \), the graph begins at the point \( (1, 0) \), since it is only defined for \( w \geq 1 \). The graph then extends to the right, forming a gentle curve upwards as \( w \) increases.

• Starting Point: \( (1, 0) \)
• Direction: Increases as \( w \) increases
• Curve: Rightward and upward opening

Sketching helps visualize the function's properties: its domain, when it's increasing or decreasing, and provides insight into its behavior over the set domain.