Question
What are the domain and range of the following function?
f(x) = x^2 + 3x − 4
f(x) = x^2 + 3x − 4
Answers
GPT 3.5
The domain of the function is all real numbers because there are no restrictions on the values of x.
To find the range of the function, we can analyze its graph or use algebraic methods. The function is a quadratic, so we know that the graph is a parabola. Since the coefficient of the x^2 term is positive, the parabola opens upwards and the range is bounded below.
To find the minimum value of the function, we can find the x-coordinate of the vertex. The x-coordinate of the vertex is given by x = -b/(2a), where a and b are the coefficients of the quadratic terms. In this case, a = 1 and b = 3, so x = -3/(2*1) = -3/2.
Substituting this value back into the function, we can find the y-coordinate of the vertex: f(-3/2) = (-3/2)^2 + 3(-3/2) - 4 = 9/4 - 9/2 - 4 = -15/4.
So, the vertex of the parabola is (-3/2, -15/4). Since the parabola opens upwards, the range of the function is all real numbers greater than or equal to -15/4.
Therefore, the range of the function is (-∞, -15/4] if written in interval notation.
To find the range of the function, we can analyze its graph or use algebraic methods. The function is a quadratic, so we know that the graph is a parabola. Since the coefficient of the x^2 term is positive, the parabola opens upwards and the range is bounded below.
To find the minimum value of the function, we can find the x-coordinate of the vertex. The x-coordinate of the vertex is given by x = -b/(2a), where a and b are the coefficients of the quadratic terms. In this case, a = 1 and b = 3, so x = -3/(2*1) = -3/2.
Substituting this value back into the function, we can find the y-coordinate of the vertex: f(-3/2) = (-3/2)^2 + 3(-3/2) - 4 = 9/4 - 9/2 - 4 = -15/4.
So, the vertex of the parabola is (-3/2, -15/4). Since the parabola opens upwards, the range of the function is all real numbers greater than or equal to -15/4.
Therefore, the range of the function is (-∞, -15/4] if written in interval notation.