Question

practice
1 graph each function. determine whether the function is even, odd, or neither. then, describe the end behavior of the graph.
ⓐ ( f(x) = x^3 + x )
ⓑ ( f(x) = x^3 + x^2 - 6x )

Explanation:

Part (a) - \( f(x) = x^3 + x \)

Step 1: Check for Even/Odd Function

To determine if a function is even or odd, we use the definitions:

• A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain.
• A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

Calculate \( f(-x) \):
\( f(-x) = (-x)^3 + (-x) = -x^3 - x \)

Now, check \( -f(x) \):
\( -f(x) = -(x^3 + x) = -x^3 - x \)

Since \( f(-x) = -f(x) \), the function is odd.

Step 2: Analyze End Behavior

For a polynomial function \( f(x) = a_nx^n + \dots + a_1x + a_0 \), the end behavior is determined by the leading term (the term with the highest degree), \( a_nx^n \).

For \( f(x) = x^3 + x \), the leading term is \( x^3 \) (degree 3, odd; leading coefficient \( 1 \), positive).

• As \( x \to +\infty \), \( x^3 \to +\infty \), so \( f(x) \to +\infty \).
• As \( x \to -\infty \), \( x^3 \to -\infty \) (since \( (-\infty)^3 = -\infty \)), so \( f(x) \to -\infty \).

Step 3: Graphical Check (from the given graph)

The graph passes through the origin, is symmetric about the origin (which confirms it's odd), and as \( x \) increases to \( +\infty \), the graph rises, and as \( x \) decreases to \( -\infty \), the graph falls, matching our end - behavior analysis.

Part (b) - \( f(x) = x^3 + x^2 - 6x \)

Step 1: Check for Even/Odd Function

Calculate \( f(-x) \):
\( f(-x)=(-x)^3+(-x)^2 - 6(-x)=-x^3 + x^2+6x \)

Now, check \( f(x) \) and \( -f(x) \):
\( f(x)=x^3 + x^2 - 6x \)
\( -f(x)=-x^3 - x^2 + 6x \)

Since \( f(-x)
eq f(x) \) (not even) and \( f(-x)
eq -f(x) \) (not odd), the function is neither even nor odd.

Step 2: Analyze End Behavior

The leading term of \( f(x)=x^3 + x^2 - 6x \) is \( x^3 \) (degree 3, odd; leading coefficient \( 1 \), positive).

• As \( x \to +\infty \), \( x^3 \to +\infty \), so \( f(x) \to +\infty \).
• As \( x \to -\infty \), \( x^3 \to -\infty \), so \( f(x) \to -\infty \).

Step 3: Graphical Check (from the given graph)

The graph is not symmetric about the y - axis (so not even) and not symmetric about the origin (so not odd), which confirms it's neither. Also, as \( x \to +\infty \), the graph rises, and as \( x \to -\infty \), the graph falls, matching the end - behavior analysis of the leading term \( x^3 \).

Answer:

s:

Part (a)

• Even/Odd: Odd
• End Behavior: As \( x\to+\infty \), \( f(x)\to+\infty \); as \( x\to-\infty \), \( f(x)\to-\infty \)

Part (b)

• Even/Odd: Neither
• End Behavior: As \( x\to+\infty \), \( f(x)\to+\infty \); as \( x\to-\infty \), \( f(x)\to-\infty \)