the quadratic function f (x) = x^2 + 2x - 15 is shown in the graph. determine the domain and range of this function.

1 answer

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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