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Determine whether the function f(x) = a^...

Determine whether the function `f(x) = a^(x) - a^(-x) + sinx` is even or odd.

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To determine whether the function \( f(x) = a^x - a^{-x} + \sin x \) is even or odd, we will follow these steps: ### Step 1: Recall the definitions of even and odd functions. - A function \( f \) is **even** if \( f(-x) = f(x) \). - A function \( f \) is **odd** if \( f(-x) = -f(x) \). ### Step 2: Calculate \( f(-x) \). We start by substituting \(-x\) into the function: \[ f(-x) = a^{-x} - a^{-(-x)} + \sin(-x) \] This simplifies to: \[ f(-x) = a^{-x} - a^{x} + \sin(-x) \] ### Step 3: Simplify \( f(-x) \). Using the property of sine, we know that \( \sin(-x) = -\sin(x) \). Therefore: \[ f(-x) = a^{-x} - a^{x} - \sin(x) \] ### Step 4: Rearrange \( f(-x) \). Now we can rearrange \( f(-x) \): \[ f(-x) = -a^{x} + a^{-x} - \sin(x) \] This can be rewritten as: \[ f(-x) = -(a^{x} - a^{-x} + \sin(x)) \] Thus: \[ f(-x) = -f(x) \] ### Step 5: Conclusion. Since we have shown that \( f(-x) = -f(x) \), we conclude that the function \( f(x) \) is **odd**. ### Summary: The function \( f(x) = a^x - a^{-x} + \sin x \) is an odd function. ---
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