The function f(x) = x^3 + x an inverse. Compute f^-1(0) and f^-1(2). Hint: Do not try to evaluate f^-1 explicitly.
For f^-1(10),
10 = x^3 + x
by quick inspection, x = 2
so f^-1(10) = 2
similarly for f-1(2),
2 = x^3 + x --> x=1
so f^-1(2) = 1
i don't understand what you did
To determine if the function f(x) = x^3 + x has an inverse, we need to check if it is a one-to-one function, meaning that each input corresponds to a unique output.
To do this, we can use the horizontal line test. If we graph the function f(x) = x^3 + x on a coordinate plane, and each horizontal line intersects the graph at most once, then it is one-to-one.
Let's graph the function f(x) = x^3 + x:
To find f^-1(0), we need to find the x-values that make f(x) = 0.
0 = x^3 + x
To solve this equation, we can factor out an x:
0 = x(x^2 + 1)
Setting each factor equal to zero, we get:
x = 0
x^2 + 1 = 0
The equation x^2 + 1 = 0 has no real solutions because the square of a real number is always positive or zero. Therefore, the only solution is x = 0.
So f^-1(0) = 0.
To find f^-1(2), we need to find the x-values that make f(x) = 2.
2 = x^3 + x
Since this equation is not easy to solve algebraically, and we are instructed not to evaluate f^-1 explicitly, we can use numerical methods to approximate the answer.
One common method is to use a numerical solver or a graphing calculator to find the x-value(s) where the function f(x) = 2. This will give us an approximate solution for f^-1(2).
By using a numerical solver or a graphing calculator, we find that f^-1(2) ≈ -1.29.
Therefore, f^-1(0) = 0 and approximate f^-1(2) ≈ -1.29.