The function \( f(x) = x^2 + 2x - 15 \) is a quadratic function, which is represented by a parabola.
Domain:
The domain of any quadratic function is all real numbers, since you can input any real number for \( x \). Therefore, the domain of \( f(x) \) is: \[ \text{Domain: } (-\infty, \infty) \]
Range:
To determine the range, we need to find the vertex of the parabola, as it opens upwards (the coefficient of \( x^2 \) is positive). The vertex form can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \).
In the function \( f(x) = x^2 + 2x - 15 \):
- \( a = 1 \)
- \( b = 2 \)
Now, we can calculate the x-coordinate of the vertex: \[ x = -\frac{2}{2(1)} = -1 \]
Next, we find the y-coordinate of the vertex by plugging \( x = -1 \) back into the function: \[ f(-1) = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16 \]
Thus, the vertex is at the point \((-1, -16)\). Since this is the minimum point of the upwards-opening parabola, the range starts from this y-value and goes to positive infinity.
Thus, the range of the function is: \[ \text{Range: } [-16, \infty) \]
Summary:
- Domain: \( (-\infty, \infty) \)
- Range: \( [-16, \infty) \)
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