2) Consider the function \( f(x)=2 x^{2} \) and \( p(x)=\left(\frac{1}{2}\right)^{x} \). a) Restrict the domain of \( f \) so that the inverse of \( f \) will also be a function. b) Write down the equation of the inverse of \( p \) in the form \( p^{-1}(x)=\ldots \) c) If the inverse of another function is \( g^{-1}(x)=2 x-4 \). Determine the equation of \( g(x) \).

Let's address each part of the problem step-by-step. --- ### **Problem 2(a)** **Restrict the domain of \( f(x) = 2x^2 \) so that its inverse is also a function.** **Solution:** The function \( f(x) = 2x^2 \) is a **quadratic function**, which is **not one-to-one** over its entire domain \( \mathbb{R} \) because it fails the **horizontal line test**. To ensure that an inverse exists and is also a function, we need to **restrict the domain** of \( f \) so that it becomes **one-to-one**. Typically, for quadratic functions, the domain is restricted to either: - \( x \geq 0 \) (the right half of the parabola), or - \( x \leq 0 \) (the left half of the parabola). **Standard Practice:** Restrict the domain to \( x \geq 0 \). **Answer:** **a)** Restrict \( f \) to all real numbers \( x \) with \( x \geq 0 \), making \( f \) one-to-one. --- ### **Problem 2(b)** **Find the inverse of \( p(x) = \left(\frac{1}{2}\right)^x \) in the form \( p^{-1}(x) = \ldots \)** **Solution:** To find the inverse function \( p^{-1}(x) \), follow these steps: 1. **Express the function with \( y \):** \[ y = \left(\frac{1}{2}\right)^x \] 2. **Take the natural logarithm of both sides:** \[ \ln y = x \ln\left(\frac{1}{2}\right) \] 3. **Solve for \( x \):** \[ x = \frac{\ln y}{\ln\left(\frac{1}{2}\right)} = \frac{\ln y}{- \ln 2} = -\frac{\ln y}{\ln 2} \] 4. **Express the inverse function:** \[ p^{-1}(x) = -\frac{\ln x}{\ln 2} \] This can also be written using the logarithm base \( \frac{1}{2} \): \[ p^{-1}(x) = \log_{\frac{1}{2}} x \] **Answer:** **b)** The inverse of \( p \) is \( p^{-1}(x) = \dfrac{\ln\,x}{\ln\,\tfrac{1}{2}} \). --- ### **Problem 2(c)** **Given that the inverse of another function is \( g^{-1}(x) = 2x - 4 \). Determine the equation of \( g(x) \).** **Solution:** To find \( g(x) \) when \( g^{-1}(x) = 2x - 4 \), we can use the property that \( g \) and \( g^{-1} \) are inverse functions. This means: \[ g(g^{-1}(x)) = x \] To find \( g(x) \), solve the equation \( y = g^{-1}(x) \) for \( x \): 1. **Start with the inverse function:** \[ y = 2x - 4 \] 2. **Solve for \( x \):** \[ y + 4 = 2x \\ x = \frac{y + 4}{2} \] 3. **Express \( g(x) \) by switching \( x \) and \( y \):** \[ g(x) = \frac{x + 4}{2} = \frac{1}{2}x + 2 \] **Answer:** **c)** The original function is \( g(x) = \dfrac{\,x\,+\,4\,}{\,2\,} \). --- If you have any further questions or need more explanations, feel free to ask!