The function, f is defined by 𝑓: 𝑥 → 2𝑥−3
𝑥
, 𝑥 ≠
0. Find 𝑓−1
A. 3
𝑥−2
B. 3
2−𝑥
C. 2
𝑥−3
D. 2
3−𝑥
Since they stipulated that 𝑥 ≠0, I assume they meant
f(x) = (2x-3)/x = 2 - 3/x
so the inverse is B: y = 3/(2-x)
You are correct, if we interpret the function as f(x) = (2x-3)/x for x ≠ 0, then the inverse function would be y = 3/(2-x), which corresponds to answer choice B. Thank you for catching that!
To find the inverse function, denoted as f^-1, we need to swap the roles of x and f(x) and solve for x.
Given the function f(x) = 2x - 3, let's replace f(x) with y:
y = 2x - 3
Now, we'll solve for x in terms of y:
y + 3 = 2x
x = (y + 3)/2
So, the inverse function, f^-1(y), is:
f^-1(y) = (y + 3)/2
To write it in terms of x, we substitute y with x:
f^-1(x) = (x + 3)/2
Therefore, the correct answer is C. 2(x - 3).
To find the inverse of the function f, you need to swap the roles of x and f(x) and solve for x.
The function f is defined as f(x) = 2x - 3, x ≠ 0.
To find f^(-1), replace f(x) with y and x with f^(-1)(y):
y = 2f^(-1)(y) - 3
Now, let's solve for f^(-1)(y):
y = 2f^(-1)(y) - 3
Add 3 to both sides:
y + 3 = 2f^(-1)(y)
Divide both sides by 2:
(y + 3) / 2 = f^(-1)(y)
Therefore, the inverse function f^(-1)(y) = (y + 3) / 2.
Substituting back x for y, the inverse function becomes:
f^(-1)(x) = (x + 3) / 2.
Therefore, the correct answer is option A: 3/(x - 2).
To find the inverse of f, we need to solve for x in terms of f(x).
Starting with f(x) = 2x - 3:
f(x) = y = 2x - 3
Add 3 to both sides:
y + 3 = 2x
Divide by 2:
x = (y + 3)/2
So the inverse function is:
f^-1(x) = (x + 3)/2
Substituting y back in for x:
f^-1(y) = (y + 3)/2
Therefore, the answer is A) 3/(x-2).