find the domain and range of 
a) f(x) = 4 / 2- x 
b) f(t) = 1 - square root t

Let's solve each part step by step to find the domain and range.

### a) \( f(x) = \frac{4}{2 - x} \)

#### **Domain:**
For the domain, we need to ensure that the denominator is not zero, because division by zero is undefined.

Set the denominator equal to zero:
\[
2 - x = 0 \quad \Rightarrow \quad x = 2
\]

Thus, the function is undefined at \( x = 2 \). Therefore, the domain is all real numbers except \( x = 2 \):
\[
\text{Domain of } f(x) = (-\infty, 2) \cup (2, \infty)
\]

#### **Range:**
To find the range, observe that the output of \( f(x) = \frac{4}{2 - x} \) can take any real value except zero, because the numerator is a constant (4), and there's no value of \( x \) that makes \( \frac{4}{2 - x} = 0 \).

Thus, the range is:
\[
\text{Range of } f(x) = (-\infty, 0) \cup (0, \infty)
\]

---

### b) \( f(t) = 1 - \sqrt{t} \)

#### **Domain:**
For the domain, we need to ensure that the expression inside the square root is non-negative, as the square root of a negative number is not defined in the set of real numbers.

So, \( t \geq 0 \).

Thus, the domain is:
\[
\text{Domain of } f(t) = [0, \infty)
\]

#### **Range:**
To find the range, note that \( \sqrt{t} \geq 0 \) for all \( t \geq 0 \), which means that \( f(t) = 1 - \sqrt{t} \leq 1 \).

As \( t \to \infty \), \( \sqrt{t} \to \infty \), and \( f(t) \to -\infty \).

Therefore, the range is:
\[
\text{Range of } f(t) = (-\infty, 1]
\]

---

### Summary:
- **a)**
  - Domain: \( (-\infty, 2) \cup (2, \infty) \)
  - Range: \( (-\infty, 0) \cup (0, \infty) \)
- **b)**
  - Domain: \( [0, \infty) \)
  - Range: \( (-\infty, 1] \)

---

Would you like more details or have any questions about these calculations?

#### 5 Related Questions:
1. What happens to the graph of \( f(x) = \frac{4}{2 - x} \) as \( x \) approaches 2?
2. How can we modify the function \( f(t) = 1 - \sqrt{t} \) to change its domain?
3. What are vertical and horizontal asymptotes, and how do they apply to rational functions like \( f(x) = \frac{4}{2 - x} \)?
4. How do we handle domain restrictions for more complex functions involving both square roots and fractions?
5. Can we determine the range of a function graphically without doing algebraic calculations?

#### Tip:
When dealing with square roots, always remember that the expression inside must be non-negative to ensure the function is defined in the real number system.

find the domain and range of 
a) f(x) = 4 / 2 - x 
b) f(t) = 1 - square root t

Learn how to find the domain and range of the functions f(x) = 4 / (2 - x) and f(t) = 1 - √t with detailed step-by-step solutions. This problem covers key algebraic concepts such as rational functions and square roots. Suitable for students in Grades 9-12.

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