Final answer:
The inverse of the function f(x) = -3x+5 can be found by switching the x and y variables and solving for y. For each statement, option A is not true because the domain of f-1(2) is a single point, not an interval. Option B is true because the inverse function has a constant rate of change. Option C is not true because the x-intercept of f(x) is (5/3, 0), not (0, 0). Option D is true because the range of the inverse function is (-∞,∞).
Step-by-step explanation:
The inverse of the function f(x) = -3x+5 can be found by switching the x and y variables and solving for y. So, let's start:
- Swap the x and y: x = -3y+5
- Solve for y: -3y = x-5
- Divide by -3 on both sides: y = (x-5)/-3 = 5/3 - x/3
Now, let's address each statement:
A. The domain of f-1(2) is (-∞,0): Since the inverse function finds the y-values for a given x-value, the statement is not true. The domain of f-1(2) is the set of x-values that make the expression (x-5)/-3 = 2 true, which is a single point, not an interval.
B. f-1(x) has a constant rate of change: This statement is true. The inverse function f-1(x) is a straight line with a constant rate of change because its equation is linear.
C. The x-intercept of f(x) is (0,0): Since the original function f(x) = -3x+5 is a straight line, its x-intercept is the point where y is equal to 0. By substituting y = 0 in the equation and solving for x, we find that the x-intercept is (5/3, 0), not (0, 0).
D. The range of f-1(x) is (-∞,∞): Since the inverse function f-1(x) is a straight line with a slope of -1/3, its range is (-∞,∞), meaning it takes on all possible y-values.