## Problem Given: \( f(x) = x^4 \) **Question:** Why does \( f(x) \) have NO inverse function? --- ## Step-by-Step Explanation ### 1. **Definition of an Inverse Function** A function \( f \) has an inverse if and only if it is **one-to-one** (injective). This means: > For every \( a \) and \( b \) in the domain, if \( f(a) = f(b) \), then \( a = b \). --- ### 2. **Check if \( f(x) = x^4 \) is One-to-One** Let’s test with two different values: - Let \( x_1 = 2 \) - Let \( x_2 = -2 \) Calculate: - \( f(2) = 2^4 = 16 \) - \( f(-2) = (-2)^4 = 16 \) But \( 2 \neq -2 \). So, **two different inputs give the same output**. --- ### 3. **Conclusion** Since \( f(x) = x^4 \) is **not one-to-one** over all real numbers, **it does not have an inverse function** (over \( \mathbb{R} \)). --- ## **Final Answer** > **\( f(x) = x^4 \) does not have an inverse function because it is not one-to-one (injective). For example, \( f(2) = f(-2) \) but \( 2 \neq -2 \).**