Question

Sketch the graphs of f(x)=5x5−3x
and g(x)=−f(−x)
. Based on the graphs, is f(x)
an even function, an odd function, or neither even nor odd?

Option 1: even function

Option 2: odd function

Option 3: neither even nor odd function

(1 point)
The option that describes the function f(x)
is Option

Answer 1

To determine whether the function \( f(x) = 5x^5 - 3x \) is even, odd, or neither, we can use the definitions of even and odd functions:

- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 1: Calculate \( f(-x) \)
Let's find \( f(-x) \):

\[
f(-x) = 5(-x)^5 - 3(-x)
\]

Calculating each term:

\[
(-x)^5 = -x^5 \quad \text{(since the exponent is odd)}
\]
\[
f(-x) = 5(-x^5) + 3x = -5x^5 + 3x
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \)

- We know \( f(x) = 5x^5 - 3x \).

Now, let's check if \( f(-x) = f(x) \) (for even) or \( f(-x) = -f(x) \) (for odd):

1. **Check for even function:**
\[
f(-x) = -5x^5 + 3x \quad \text{(not equal to \( 5x^5 - 3x \))}
\]

2. **Check for odd function:**
\[
-f(x) = -(5x^5 - 3x) = -5x^5 + 3x
\]
\[
f(-x) = -5x^5 + 3x \quad \text{(equal to \( -f(x) \))}
\]

### Conclusion
Since \( f(-x) = -f(x) \), the function \( f(x) = 5x^5 - 3x \) is an **odd function**.

Thus, the option that describes the function \( f(x) \) is:

**Option 2: odd function**.