which of the following functions have a inverse?
f(x)=ab.solute value of x.
f(x)=x^3
f(x)=x^4
f(x)=x^3
f(x)=x^4,xless then or equal 0.
how do i know if they have an inverser, can someone please explain:) thanks for your time YOOO
x^3 has an inverse, as does x^4 x<=0
Why? The others do not have a one for one mapping from x to one f(x).
for instance, x^4 f(-sqrt2)=f(sqrt2)
http://www.uncwil.edu/courses/mat111hb/functions/inverse/inverse.html#sec1
To determine whether a function has an inverse, you need to check if the function satisfies the necessary conditions for an inverse function. The main condition is that the function must be one-to-one or bijective.
A function is one-to-one if each element in the domain is paired with a unique element in the range. In other words, no two different x-values can produce the same y-value.
Let's analyze each given function:
1. f(x) = ab |x| (absolute value of x):
This function does not have an inverse because it fails the one-to-one criteria. For example, if a = 1 and b = 1, both positive and negative x-values will map to the same positive y-value.
2. f(x) = x^3:
This function has an inverse because it passes the one-to-one test. Each distinct x-value produces a unique y-value. To find the inverse, you can solve for x. Start by setting y = f(x) and then solve for x. The resulting inverse function will be the cube root of y, denoted as f^(-1)(x) = ∛x.
3. f(x) = x^4:
Similar to the second function, this function also has an inverse because it is one-to-one. Each x-value corresponds to a unique y-value. To find the inverse, set y = f(x) and solve for x. The resulting inverse function will be the fourth root of y, denoted as f^(-1)(x) = ∜x.
4. f(x) = x^3, x ≤ 0:
This function does not have an inverse. Although the function is one-to-one for x-values less than or equal to zero, it fails for x-values greater than zero. This restriction means that it does not pass the one-to-one test for the entire domain, so an inverse cannot be defined.
In summary, functions 2 and 3 (f(x) = x^3 and f(x) = x^4) have inverses, while functions 1 and 4 (f(x) = ab |x| and f(x) = x^3, x ≤ 0) do not have inverses.