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 \( f \).

Sure, let's address each of the questions you've provided step by step. --- ### **Given:** - \( f(x) = \frac{1}{2} x^{2} \) - \( g(x) = 2x + 4 \) --- ### **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. --- ### **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:** \[ \text{Domain of } f(x): x \geq 0 \] --- ### **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 \( y = \ldots \):** \[ y = \frac{x - 4}{2} \] --- ### **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:** \[ \text{Range of } f(x): y \geq 0 \] --- ### **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 \( a = 1.25 \).** --- If you have any more questions or need further clarification on any of these points, feel free to ask!