Question

Find the domain and the range of the following functions
A. fx)=2x+1
B. g(x)=x³
C. h(x)=x÷x
D. t(x)=square root ²_4
E. S(x)=/x/
F. f(x)=square root x_2

Answer 1

Let's determine the domain and range for each of the functions you've provided:

### A. \( f(x) = 2x + 1 \)

- **Domain**: The function is a linear polynomial, which means it is defined for all real numbers. Thus, the domain is:
\[
\text{Domain of } f: (-\infty, \infty)
\]

- **Range**: Since it is a linear function, it can take any real value as \( x \) varies across all real numbers. Thus, the range is:
\[
\text{Range of } f: (-\infty, \infty)
\]

---

### B. \( g(x) = x^3 \)

- **Domain**: The cubic function is defined for all real numbers, so the domain is:
\[
\text{Domain of } g: (-\infty, \infty)
\]

- **Range**: Similar to linear functions, the cubic function can also take any real value. Thus, the range is:
\[
\text{Range of } g: (-\infty, \infty)
\]

---

### C. \( h(x) = \frac{x}{x} \)

- **Domain**: This function is defined for all \( x \) except \( x = 0 \) (since division by zero is undefined). Thus, the domain is:
\[
\text{Domain of } h: (-\infty, 0) \cup (0, \infty)
\]

- **Range**: For all \( x \neq 0 \), \( h(x) = 1 \). Therefore, the range is:
\[
\text{Range of } h: \{ 1 \}
\]

---

### D. \( t(x) = \sqrt{4} \)

- **Domain**: This function is a constant function since \( \sqrt{4} = 2 \). It is defined for all real numbers. Thus, the domain is:
\[
\text{Domain of } t: (-\infty, \infty)
\]

- **Range**: The function only outputs a single value (2), so the range is:
\[
\text{Range of } t: \{ 2 \}
\]

---

### E. \( S(x) = |x| \)

- **Domain**: The absolute value function is defined for all real numbers, so the domain is:
\[
\text{Domain of } S: (-\infty, \infty)
\]

- **Range**: The output of the absolute value function is always non-negative. Thus, the range is:
\[
\text{Range of } S: [0, \infty)
\]

---

### F. \( f(x) = \sqrt{x^2} \)

- **Domain**: The square root function requires that its argument be non-negative. Since \( x^2 \) is always non-negative, the function is defined for all real \( x \). Thus, the domain is:
\[
\text{Domain of } f: (-\infty, \infty)
\]

- **Range**: The output of \( \sqrt{x^2} \) is the absolute value of \( x \), which is also non-negative. Thus, the range is:
\[
\text{Range of } f: [0, \infty)
\]

In summary, here are the domains and ranges:

- **A**: Domain: \((-∞, ∞)\), Range: \((-∞, ∞)\)
- **B**: Domain: \((-∞, ∞)\), Range: \((-∞, ∞)\)
- **C**: Domain: \((-∞, 0) \cup (0, ∞)\), Range: \(\{ 1 \}\)
- **D**: Domain: \((-∞, ∞)\), Range: \(\{ 2 \}\)
- **E**: Domain: \((-∞, ∞)\), Range: \([0, ∞)\)
- **F**: Domain: \((-∞, ∞)\), Range: \([0, ∞)\)