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 $$ g(x) $$.

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Let's address each part of the problem step-by-step.

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### **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.

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### **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 $$ p^{-1}(x) = \dfrac{\ln\,x}{\ln\,\tfrac{1}{2}} $$.

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### **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 $$ g(x) = \dfrac{\,x\,+\,4\,}{\,2\,} $$.

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If you have any further questions or need more explanations, feel free to ask!
**a)** Restrict $$ f(x) = 2x^2 $$ to $$ x \geq 0 $$. **b)** The inverse of $$ p(x) = \left(\frac{1}{2}\right)^x $$ is $$ p^{-1}(x) = \dfrac{\ln\,x}{\ln\,\tfrac{1}{2}} $$. **c)** The function $$ g(x) $$ is $$ g(x) = \dfrac{x + 4}{2} $$.

Consider the function and .
a) Restrict the domain of so that the inverse of will also be a function.
b) Write down the equation of the inverse of in the form
c) If the inverse of another function is . Determine the equation of .

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Consider the function and . a) Restrict the domain of so that the inverse of will also be a function. b) Write down the equation of the inverse of in the form c) If the inverse of another function is . Determine the equation of .

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Consider the function and .
a) Restrict the domain of so that the inverse of will also be a function.
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