4. The function $$ f $$ and $$ g $$ are defined as $$ f(x)=4-x^{2}, x \in I R \quad g(x)=\sqrt{x}, x \geq 0 $$ a. State the range of $$ f $$ b. Find the domain of $$ g f(x) $$ c. Sketch the graph of $$ y=|f(x)| $$ indicating the intercepts with the $$ x $$-axis d. State with a reason, whether $$ f(x) $$ is injective or not

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**d. Determining Whether $$ f(x) = 4 - x^2 $$ is Injective**

A function is **injective** (or **one-to-one**) if every element of the function's range is mapped by exactly one element of its domain. In other words, for $$ f $$ to be injective, whenever $$ f(x_1) = f(x_2) $$, it must follow that $$ x_1 = x_2 $$.

Let's analyze the function $$ f(x) = 4 - x^2 $$:

1. **Symmetry Analysis**:
   - The function $$ f(x) = 4 - x^2 $$ is a **symmetric** function about the y-axis because $$ f(x) = f(-x) $$ for all $$ x \in \mathbb{R} $$.
   
2. **Testing Injectivity**:
   - Consider two different inputs: $$ x_1 = a $$ and $$ x_2 = -a $$ where $$ a 
eq 0 $$.
   - Compute $$ f(x_1) = 4 - a^2 $$ and $$ f(x_2) = 4 - (-a)^2 = 4 - a^2 $$.
   - Here, $$ f(x_1) = f(x_2) $$ even though $$ x_1 
eq x_2 $$.

3. **Conclusion**:
   - Since there exist distinct values $$ x_1 $$ and $$ x_2 $$ such that $$ f(x_1) = f(x_2) $$, the function $$ f(x) $$ is **not injective**.

**Reason**: The function $$ f(x) = 4 - x^2 $$ is symmetric about the y-axis, meaning that $$ f(a) = f(-a) $$ for any $$ a $$. Therefore, distinct inputs $$ a $$ and $$ -a $$ produce the same output, violating the condition for injectivity.

**Answer Summary:**

Problem 4.d Answer:

The function $$ f(x) = 4 - x^2 $$ is not injective because $$ f(a) = f(-a) $$ for any real number $$ a $$, meaning distinct inputs can produce the same output.
The function $$ f(x) = 4 - x^2 $$ is not injective because different inputs can produce the same output.

The function and are defined as

a. State the range of
b. Find the domain of
c. Sketch the graph of indicating the intercepts with the -axis
d. State with a reason, whether is injective or not

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The function and are defined as a. State the range of b. Find the domain of c. Sketch the graph of indicating the intercepts with the -axis d. State with a reason, whether is injective or not