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\( f(x)=\frac{2 x^{5}-4 x^{3}+4 x}{4 x^{3}-9 x} \) Part: \( \mathbf{0} / 2 \) Part 1 of 2 (a) Determine if the function is even, odd, or neither. The function is odd. The function is neither odd nor even. The function is even.

Ask by Ward Burgess. in the United States
Mar 18,2025

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The function is even.

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We start with the function \[ f(x)=\frac{2x^5-4x^3+4x}{4x^3-9x}. \] To determine whether \( f \) is even, odd, or neither, we calculate \( f(-x) \) and compare it with \( f(x) \). **Step 1. Substitute \(-x\) into the function.** The function becomes \[ f(-x)=\frac{2(-x)^5-4(-x)^3+4(-x)}{4(-x)^3-9(-x)}. \] **Step 2. Simplify the numerator.** Calculate each term: \[ 2(-x)^5=2(-x^5)=-2x^5, \] \[ -4(-x)^3=-4(-x^3)=4x^3, \] \[ 4(-x)=-4x. \] Thus, the numerator is \[ -2x^5+4x^3-4x. \] **Step 3. Simplify the denominator.** Calculate each term: \[ 4(-x)^3=4(-x^3)=-4x^3, \] \[ -9(-x)=9x. \] So the denominator is \[ -4x^3+9x. \] Thus, we have \[ f(-x)=\frac{-2x^5+4x^3-4x}{-4x^3+9x}. \] **Step 4. Factor out \(-1\) from the numerator and the denominator.** The numerator factorization: \[ -2x^5+4x^3-4x = -\left(2x^5-4x^3+4x\right). \] The denominator factorization: \[ -4x^3+9x = -\left(4x^3-9x\right). \] So, the function becomes \[ f(-x)=\frac{-\left(2x^5-4x^3+4x\right)}{-\left(4x^3-9x\right)}. \] **Step 5. Cancel out the common factor \(-1\).** Cancelling the \(-1\) from the numerator and denominator, we obtain \[ f(-x)=\frac{2x^5-4x^3+4x}{4x^3-9x}=f(x). \] **Conclusion:** Since \( f(-x)=f(x) \) for all \( x \) in the domain of \( f \), the function \( f \) is even. **Answer:** \[ \text{The function is even.} \]

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Bonus Knowledge

To determine if the function \( f(x)=\frac{2 x^{5}-4 x^{3}+4 x}{4 x^{3}-9 x} \) is even, odd, or neither, we can evaluate \( f(-x) \). Calculating \( f(-x) \): \[ f(-x) = \frac{2 (-x)^{5} - 4 (-x)^{3} + 4 (-x)}{4 (-x)^{3} - 9 (-x)} = \frac{-2 x^{5} + 4 x^{3} - 4 x}{-4 x^{3} + 9 x} = \frac{-2 x^{5} + 4 x^{3} - 4 x}{-4 x^{3} + 9 x} \] Now, if \( f(-x) = -f(x) \), the function would be odd. Calculating \( -f(x) \): \[ -f(x) = -\frac{2 x^{5}-4 x^{3}+4 x}{4 x^{3}-9 x} = \frac{-2 x^{5}+4 x^{3}-4 x}{4 x^{3}-9 x} \] Thus, we see that \( f(-x) = -f(x) \), confirming that the function is odd. **Final Answer: The function is odd.**

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