Find the domain and range of the function.
1. f(x)= x^2-x
I got: Domain= All Real Numbers
Range= All Real Numbers > -1 (with a line under the greater than sign)
2. f(x)= 1-x^2
I got: Domain= All Real Numbers
Range= All Real Numbers < 1
(with a line under the less than sign)
Are these right? Thanks.
The second answer is correct, but the lowest possible value (Range lower limit) of x^2-x
is f(x) = -1/4, and occurs when x = 1/2
To find the domain and range of a function, you need to consider the possible input values (domain) and the corresponding output values (range).
1. For the function f(x) = x^2 - x:
To determine the domain, you need to find the set of all possible input values (x) for which the function is defined. In this case, there are no restrictions on the input values of x. The function is defined for all real numbers, so the domain is indeed all real numbers.
To find the range, you need to determine the set of all possible output values (f(x)) for the given input values. One approach is to analyze the behavior of the function. In this case, the function is a quadratic function, which opens upward since the coefficient of x^2 is positive. This means that the vertex of the parabola represents the lowest point of the graph, and there is no upper limit for the output values.
To find the vertex of the parabola, you can use the formula x = -b/(2a), where a and b are the coefficients of x^2 and x, respectively. In this case, a = 1 and b = -1. Substituting these values into the formula, you get x = -(-1)/(2*1) = 1/2.
Substitute this x-value back into the function f(x) = x^2 - x to find the corresponding output value:
f(1/2) = (1/2)^2 - (1/2) = 1/4 - 1/2 = -1/4
Therefore, the range of the function is all real numbers greater than or equal to -1/4. In mathematical notation, we can write it as Range = ( -1/4, ∞) or with a line under the greater than sign to indicate it includes -1/4 (greater than or equal to).
Therefore, the correct answer is:
Domain = All real numbers
Range = All real numbers greater than or equal to -1/4 (with a line under the greater than sign)
2. For the function f(x) = 1 - x^2:
The domain of this function is also all real numbers, as there are no restrictions on the input values x.
To find the range, you can again analyze the behavior of the function. This function is a quadratic function that opens downward since the coefficient of x^2 is negative. This means that the vertex represents the highest point on the graph, and there is no upper limit for the output values.
To find the vertex, you can use the same formula as before: x = -b/(2a). In this case, a = -1 and b = 0. Substituting these values into the formula, you get x = -0/(2*-1) = 0.
Substituting this x-value back into the function f(x) = 1 - x^2 to find the corresponding output value:
f(0) = 1 - (0)^2 = 1 - 0 = 1
Therefore, the range of the function is all real numbers less than or equal to 1. In mathematical notation, we can write it as Range = ( -∞, 1) or with a line under the less than sign to indicate it includes 1 (less than or equal to).
Therefore, the correct answer is:
Domain = All real numbers
Range = All real numbers less than or equal to 1 (with a line under the less than sign)