8. Use set notation to describe the domain and range for $$ f(x)=x^{2}+5 $$. Hint: you can always graph a function first to help you see the domain and range more clearly.

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To determine the domain and range of the function $$ f(x) = x^2 + 5 $$, we can analyze the function step by step.

### Step 1: Identify the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

For the function $$ f(x) = x^2 + 5 $$:
- The expression $$ x^2 $$ is defined for all real numbers.
- Therefore, there are no restrictions on the values of $$ x $$.

Thus, the domain of $$ f(x) $$ is:
$$
\text{Domain} = \{ x \in \mathbb{R} \}
$$

### Step 2: Identify the Range

The range of a function is the set of all possible output values (y-values) that the function can produce.

For the function $$ f(x) = x^2 + 5 $$:
- The term $$ x^2 $$ is always non-negative (i.e., $$ x^2 \geq 0 $$).
- Therefore, the minimum value of $$ f(x) $$ occurs when $$ x^2 = 0 $$, which gives $$ f(0) = 0 + 5 = 5 $$.
- As $$ x $$ increases or decreases, $$ x^2 $$ increases, and thus $$ f(x) $$ can take on any value greater than or equal to 5.

Thus, the range of $$ f(x) $$ is:
$$
\text{Range} = \{ y \in \mathbb{R} \mid y \geq 5 \}
$$

### Conclusion

In set notation, we can summarize the domain and range of the function $$ f(x) = x^2 + 5 $$ as follows:

- **Domain**: $$ \{ x \in \mathbb{R} \} $$
- **Range**: $$ \{ y \in \mathbb{R} \mid y \geq 5 \} $$
The domain is all real numbers, and the range is all real numbers greater than or equal to 5.

8. Use set notation to describe the domain and range for \( f(x)=x^{2}+5 \). Hint: you can always graph a function first to help you see the domain and range more clearly.

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