To find the domain and range of the function \( f(x) = x^2 + 2 \), we'll analyze each aspect separately.
Domain:
The domain of a function consists of all the possible input values (x-values) for which the function is defined.
For the function \( f(x) = x^2 + 2 \):
- The expression \( x^2 \) is defined for all real numbers.
- Since there are no restrictions (like square roots of negative numbers or division by zero), the domain is all real numbers.
Domain: \( (-\infty, \infty) \)
Range:
The range of a function is the set of all possible output values (y-values) when we apply the function to its domain.
For \( f(x) = x^2 + 2 \):
- The term \( x^2 \) reaches its minimum value at \( x = 0\), which is \( 0 \).
- Thus, the minimum value of \( f(x) \) is \( 0 + 2 = 2 \).
- As \( x \) increases or decreases without bound (goes to positive or negative infinity), \( x^2 \) increases without bound, and therefore \( f(x) \) also increases without bound.
Thus, the lowest value of \( f(x) \) is \( 2 \), and it can go up to infinity.
Range: \( [2, \infty) \)
Summary:
- Domain: \( (-\infty, \infty) \)
- Range: \( [2, \infty) \)
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