Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\sqrt{1-x}\)

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt{1-x}\) is neither even nor odd because the graph does not have symmetry about y-axis or the origin.

Step 1: Determine the domain of the function

The function is \(f(x)=\sqrt{1-x}\). Because the square root of a negative number is not a real number, the domain of the function is x such that \(x \leq 1\). That means the function is only defined for values of x up to 1.

Step 2: Sketch the graph

Plugging in some values to get a rough sketch of the graph. When \(x=-1\) then \(f(x)= \sqrt{2}\),when \(x=0\) then \(f(x)= \sqrt{1}\),when \(x=1\) then \(f(x)= \sqrt{0}\).Plotting these points and connecting them will give a semi-circle opening to the left.

Step 3: Determine whether the function is even, odd, or neither

The function does not pass the tests for even or odd functions. Therefore, \(f(x)\) is neither even nor odd.

Key Concepts

Graph Sketching

Graph sketching is a visual representation of a mathematical function, displaying how the function behaves, changes, or approaches certain values. For the function \( f(x) = \sqrt{1-x} \), we can sketch its graph by evaluating some key points:
- At \( x = -1 \), \( f(x) = \sqrt{2} \).
- At \( x = 0 \), \( f(x) = \sqrt{1} = 1 \).
- At \( x = 1 \), \( f(x) = \sqrt{0} = 0 \).

Plotting these points helps in connecting them to form a semi-circle. Remember, the function's important property includes a radical, leading to a typical curve that gently bends.

When the graph is completed, it's found to open leftwards, ending at the point where \( x = 1 \) and \( y = 0 \). Sketching helps students understand the nature and behavior of a function beyond pure algebraic symbols.

Domain of a Function

The domain of a function describes the set of all possible input values (\( x \)) for which the function is defined. For the square root function \( f(x) = \sqrt{1-x} \), the expression inside the square root, \( 1-x \), must remain non-negative because taking the square root of a negative number doesn't result in a real number.

From the inequality \( 1-x \geq 0 \), we can solve for \( x \) to find \( x \leq 1 \). Thus, the solution set represents the domain of the function, which comprises all values of \( x \) from negative infinity up to 1, inclusive.

• This condition ensures the function outputs real and valid numbers only.
• Domains restrict graphing on a real number line, explaining why the sketch stops at certain points.

Knowing the domain avoids errors in calculation and aids in accurate graph interpretations.

Even and Odd Functions

Even and odd functions have distinctive algebraic and geometrical characteristics.
- **Even functions** have symmetry about the y-axis, mathematically described by \( f(x) = f(-x) \).- **Odd functions** are symmetric about the origin, indicated by \( f(x) = -f(-x) \).
To determine whether \( f(x) = \sqrt{1-x} \) is either of these, test for both conditions:

• Check for evenness: Apply \( f(x) = f(-x) \). This doesn't hold as \( \sqrt{1-x} eq \sqrt{1+x} \).
• Check for oddness: Compare \( f(x) \) to \( -f(-x) \). No match is found since \( \sqrt{1-x} eq -\sqrt{1+x} \).

Since neither condition is satisfied, \( f(x) = \sqrt{1-x} \) is categorized as neither even nor odd. This understanding helps in predicting function behavior upon graph reflections or shifts.