1. Find f^1 (x) for f(x) = -2 / 5x+3
I got f^-1 (x) = -2 - 3x / 5x
Is that right?
f = -2/(5x+3)
5x+3 = -2/f
x = (-3 - 2/f)/5 = (-3f-2)/5f
You are correct if you toss in some clarifying parentheses.
As written, -2 - 3x/5x = -2 - 3/5 - -13/5
In the absence of text formatting or parentheses, PEMDAS rules.
ok, i'll remember that next time
To find the inverse of a function, you typically replace f(x) with y, swap the x and y variables, and solve for y. Let's go through the steps for your function, f(x) = -2/(5x+3):
Step 1: Replace f(x) with y: y = -2/(5x+3)
Step 2: Swap x and y: x = -2/(5y+3)
Step 3: Solve for y:
Multiply both sides by (5y+3):
x(5y+3) = -2
Distribute x:
5xy + 3x = -2
Move 3x to the other side:
5xy = -2 - 3x
Divide by 5x to isolate y:
y = (-2 - 3x) / 5x
So it seems like there was a small error in your calculation. The correct inverse function is f^(-1)(x) = (-2 - 3x) / 5x.
Please note that the inverse of f(x) exists only if the function is one-to-one, meaning it passes the horizontal line test.