Step-by-step explanation:
To find the range of the function f(x) = 2x² - x, we need to determine the set of all possible values that the function can take. In other words, we need to find the set of all possible outputs (y-values) for the given domain (x-values).
First, let's rewrite the function in standard form: f(x) = 2x² - x = 2x(x - 1).
To find the range, we can analyze the behavior of the quadratic term 2x². Since the coefficient of x² is positive (2 > 0), the parabola opens upward. This means that the vertex of the parabola represents the minimum value of the function.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = 2 and b = -1. Substituting these values, we get x = -(-1) / (2 * 2) = 1/4.
Now, substitute this x-value into the function to find the corresponding y-value (minimum value): f(1/4) = 2(1/4)(1/4 - 1) = -1/8.
So, the range of f(x) is (-∞, -1/8].
Next, let's find the domain and range of the composite functions:
i) fg(x) = f(g(x))
To find the domain of fg(x), we need to consider the domains of both f(x) and g(x). Since both f(x) and g(x) are defined for all real numbers, the domain of fg(x) is also all real numbers, denoted as x ∈ ℝ.
To find the range of fg(x), we need to substitute the range of g(x) into f(x). However, the range of g(x) is not provided in the question. Please provide the range of g(x) so that we can proceed with finding the range of fg(x).
ii) gf(x) = g(f(x))
To find the domain of gf(x), we need to consider the domains of both g(x) and f(x). Since both g(x) and f(x) are defined for all real numbers, the domain of gf(x) is also all real numbers, denoted as x ∈ ℝ.
To find the range of gf(x), we need to substitute the range of f(x) into g(x). From the previous analysis, we found that the range of f(x) is (-∞, -1/8]. However, the range of g(x) is not provided in the question. Please provide the range of g(x) so that we can proceed with finding the range of gf(x).
