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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

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
yesterday

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.