Question
What are the domain and range of the following function?
f(x) = x^2 + 3x − 4
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f(x) = x^2 + 3x − 4
show all your steps
Answers
GPT 3.5
To find the domain of the function, we need to identify any values of x that would make the function undefined. Since there are no restrictions or limitations on the x values in the given function, the domain is all real numbers.
To find the range of the function, we need to determine all the possible output values (y-values) of the function. We can do this by analyzing the graph of the function or by observing the behavior of the function.
Since the given function is a quadratic function f(x) = x^2 + 3x − 4, we know that its graph is a parabola. The coefficient of the x^2 term is positive, which means that the parabola opens upwards. This implies that the range of the function is all real numbers greater than or equal to the vertex of the parabola.
To find the vertex of the parabola, we can use the formula x = -b/2a, where a is the coefficient of the x^2 term and b is the coefficient of the x term.
For the given function, a = 1 and b = 3. Therefore, x = -(3)/(2*1) = -3/2.
To find the corresponding y-value, we substitute the x-value of the vertex into the function:
f(-3/2) = (-3/2)^2 + 3(-3/2) - 4 = 9/4 - 9/2 - 4 = -17/4.
Thus, the vertex of the parabola (and the minimum point of the function) is (-3/2, -17/4).
Therefore, the range of the function f(x) = x^2 + 3x − 4 is all real numbers greater than or equal to -17/4. In interval notation, the range can be written as [-17/4, +∞).
To find the range of the function, we need to determine all the possible output values (y-values) of the function. We can do this by analyzing the graph of the function or by observing the behavior of the function.
Since the given function is a quadratic function f(x) = x^2 + 3x − 4, we know that its graph is a parabola. The coefficient of the x^2 term is positive, which means that the parabola opens upwards. This implies that the range of the function is all real numbers greater than or equal to the vertex of the parabola.
To find the vertex of the parabola, we can use the formula x = -b/2a, where a is the coefficient of the x^2 term and b is the coefficient of the x term.
For the given function, a = 1 and b = 3. Therefore, x = -(3)/(2*1) = -3/2.
To find the corresponding y-value, we substitute the x-value of the vertex into the function:
f(-3/2) = (-3/2)^2 + 3(-3/2) - 4 = 9/4 - 9/2 - 4 = -17/4.
Thus, the vertex of the parabola (and the minimum point of the function) is (-3/2, -17/4).
Therefore, the range of the function f(x) = x^2 + 3x − 4 is all real numbers greater than or equal to -17/4. In interval notation, the range can be written as [-17/4, +∞).