To find the domain and range of the function $f(x) = x^2 + 4$:

1. Domain:

The domain of a function refers to the set of all possible input values (x-values) that the function can accept.

Since the function is a polynomial (a quadratic function), and polynomials are defined for all real numbers, there are no restrictions on $x$. Therefore, the domain of $f(x)$ is:

$\boxed{\text{Domain: } (-\infty, \infty)}$

2. Range:

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

For $f(x) = x^2 + 4$, the smallest value of $x^2$ is 0 (since $x^2 \geq 0$ for all real numbers), and the smallest value of the function occurs when $x = 0$. At that point:

$f(0) = 0^2 + 4 = 4$

Since $x^2$ can grow larger as $x$ increases or decreases, the function will take all values greater than or equal to 4. Therefore, the range of the function is:

$\boxed{\text{Range: } [4, \infty)}$

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Would you like a more detailed explanation or have further questions?

5 Related Questions:

1. How do you determine the domain and range for other types of functions (like rational or logarithmic)?
2. What is the shape of the graph of $f(x) = x^2 + 4$?
3. How do transformations affect the domain and range of a quadratic function?
4. How would the domain and range change if the function were $f(x) = \sqrt{x^2 + 4}$?
5. What is the vertex of the quadratic function $f(x) = x^2 + 4$?

Tip: The minimum or maximum value of a quadratic function $f(x) = ax^2 + bx + c$ can always be found at $x = -\frac{b}{2a}$, and it helps in determining the range quickly.