Let's analyze the given functions and determine their domains and ranges step by step.

1. Function: $f(x) = \sqrt{x}$

Domain:

The square root function is defined only when the value inside the square root is non-negative ($x \geq 0$).

• Therefore, the domain of $f(x)$ is: $\text{Domain of } f(x): [0, \infty)$

Range:

The square root function produces non-negative values only ($\sqrt{x} \geq 0$).

• Therefore, the range of $f(x)$ is: $\text{Range of } f(x): [0, \infty)$

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2. Function: $g(x) = 1 + x$

Domain:

The function $g(x)$ is a linear equation, which is defined for all real numbers ($x \in \mathbb{R}$).

• Therefore, the domain of $g(x)$ is: $\text{Domain of } g(x): (-\infty, \infty)$

Range:

Since $g(x) = 1 + x$ is a linear function, it can take any real value depending on $x$. Thus, the range is: $\text{Range of } g(x): (-\infty, \infty)$

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Final Answer:

• Domain of $f(x)$: $[0, \infty)$
• Range of $f(x)$: $[0, \infty)$
• Domain of $g(x)$: $(-\infty, \infty)$
• Range of $g(x)$: $(-\infty, \infty)$

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Would you like a graph to visualize these domains and ranges, or any further clarification?

Related Questions:

1. What happens to the range of $f(x)$ if it is modified to $f(x) = \sqrt{x+2}$?
2. How do the domains of $f(x)$ and $g(x)$ change if we restrict $x$ to integers only?
3. What is the composition of $f(g(x))$ and its domain and range?
4. Can you define $g(f(x))$ and determine its domain and range?
5. How would adding constraints like $x > 2$ change the domain and range of these functions?

Tip: Always check the argument of square roots and denominators when determining the domain of a function!