4.6 Indicate one wayction. Given: $$ f(x)=\frac{1}{2} x^{2} $$ and $$ g(x)=2 x+4 $$ 4.1 Is $$ f(x) $$ a function? Give a reason for your answer. 4.2 Write down the domain of $$ f(x) $$ for $$ f^{-1}(x) $$ to be a function. 4.3 Write down the equation of $$ g^{-1}(x) $$ in the form $$ y=\ldots $$ 4.4 Write down the range of $$ f(x) $$. 5 The graph of $$ f(x)=a x^{2}, x \leq 0 $$ is sketched alongside. The point $$ A(-2 ; 5) $$ lies on $$ f $$.

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Sure, let's address each of the questions you've provided step by step.

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### **Given:**
- $$ f(x) = \frac{1}{2} x^{2} $$
- $$ g(x) = 2x + 4 $$

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### **4.1. Is $$ f(x) $$ a function? Give a reason for your answer.**

**Answer:**

Yes, $$ f(x) = \frac{1}{2} x^{2} $$ is a function.

**Reason:**
A relation is considered a function if every input $$ x $$ has exactly one output $$ f(x) $$. In this case, for every real number $$ x $$, squaring it and then multiplying by $$ \frac{1}{2} $$ will yield one unique output. Therefore, $$ f(x) $$ satisfies the definition of a function.

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### **4.2. Write down the domain of $$ f(x) $$ for $$ f^{-1}(x) $$ to be a function.**

**Answer:**

To have an inverse function $$ f^{-1}(x) $$, the original function $$ f(x) $$ must be **injective** (one-to-one). The function $$ f(x) = \frac{1}{2} x^{2} $$ is a parabola, which is not one-to-one over its entire domain because both positive and negative values of $$ x $$ produce the same $$ f(x) $$.

**Restricting the Domain:**
- **Option 1:** $$ x \geq 0 $$
- **Option 2:** $$ x \leq 0 $$

By restricting $$ x $$ to non-negative or non-positive values, $$ f(x) $$ becomes one-to-one, ensuring that $$ f^{-1}(x) $$ is also a function.

**Example Domain Restriction:**
$$ \text{Domain of } f(x): x \geq 0 $$

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### **4.3. Write down the equation of $$ g^{-1}(x) $$ in the form $$ y = \ldots $$**

**Answer:**

To find the inverse function $$ g^{-1}(x) $$ of $$ g(x) = 2x + 4 $$, follow these steps:

1. **Start with the original function:**
   $$ y = 2x + 4 $$

2. **Solve for $$ x $$ in terms of $$ y $$:**
   $$ y - 4 = 2x $$
   $$ x = \frac{y - 4}{2} $$

3. **Express the inverse function:**
   $$ g^{-1}(x) = \frac{x - 4}{2} $$
   
   **Or in the form $$ y = \ldots $$:**
   $$ y = \frac{x - 4}{2} $$

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### **4.4. Write down the range of $$ f(x) $$.**

**Answer:**

The function $$ f(x) = \frac{1}{2} x^{2} $$ is a quadratic function that opens upwards.

- **Minimum Value:** When $$ x = 0 $$, $$ f(0) = 0 $$.
- **Behavior:** As $$ |x| $$ increases, $$ f(x) $$ increases without bound.

**Therefore, the range of $$ f(x) $$ is:**
$$ \text{Range of } f(x): y \geq 0 $$

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### **5. The graph of $$ f(x) = a x^{2} $$, $$ x \leq 0 $$ is sketched alongside. The point $$ A(-2; 5) $$ lies on $$ f $$.**

**Question (Assumed):** Find the value of $$ a $$.

**Answer:**

Given the point $$ A(-2, 5) $$ lies on the graph of $$ f(x) = a x^{2} $$, we can substitute $$ x = -2 $$ and $$ f(x) = 5 $$ into the equation to find $$ a $$.

1. **Substitute the values:**
   $$ 5 = a (-2)^{2} $$
   $$ 5 = a \times 4 $$

2. **Solve for $$ a $$:**
   $$ a = \frac{5}{4} $$
   $$ a = 1.25 $$

**Therefore, $$ a = \frac{5}{4} $$ or $$ a = 1.25 $$.**

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If you have any more questions or need further clarification on any of these points, feel free to ask!
**4.1. Yes, $$ f(x) = \frac{1}{2} x^{2} $$ is a function because for every $$ x $$, there's exactly one $$ f(x) $$.** **4.2. To have an inverse function, restrict $$ x $$ to $$ x \geq 0 $$.** **4.3. The inverse of $$ g(x) = 2x + 4 $$ is $$ g^{-1}(x) = \frac{x - 4}{2} $$.** **4.4. The range of $$ f(x) = \frac{1}{2} x^{2} $$ is $$ y \geq 0 $$.** **5. The value of $$ a $$ is $$ \frac{5}{4} $$ or $$ 1.25 $$.**

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