Given f(x)=2x^2-1, determine the equation of the inverse.
Interchange x and y, and solve for y in terms of x.
y=2x²-1
x=2y²-1
y²=(x+1)/2
y=√((x+1)/2)
Check:
f(√((x+1)/2))
=2(x+1)/2 -1
=x+1-1
=x OK
Well, to find the inverse of f(x) = 2x^2 - 1, we'll have to do a bit of mathematical magic. First, let's replace f(x) with y:
y = 2x^2 - 1
Next, let's swap x and y:
x = 2y^2 - 1
Now, our goal is to solve for y. So, let's rearrange the equation:
2y^2 = x + 1
Divide both sides by 2:
y^2 = (x + 1)/2
To eliminate the square, we take the square root of both sides (don't worry, nothing spooky will happen):
y = ±√[(x + 1)/2]
Now, we could have two potential inverses: y = √[(x + 1)/2] and y = -√[(x + 1)/2]. But let's face it, we can't have negative clowns running around, so we'll choose the positive square root:
y = √[(x + 1)/2]
Voila! The equation of the inverse is y = √[(x + 1)/2].
To find the equation of the inverse of a function, we can follow these steps:
Step 1: Replace f(x) with y:
y = 2x^2 - 1
Step 2: Swap x and y:
x = 2y^2 - 1
Step 3: Solve the equation for y:
x + 1 = 2y^2
(x + 1) / 2 = y^2
√((x + 1) / 2) = y
Step 4: Replace y with f^(-1)(x):
f^(-1)(x) = √((x + 1) / 2)
Therefore, the equation of the inverse function is f^(-1)(x) = √((x + 1) / 2).
To find the equation of the inverse of a function, you need to follow these steps:
Step 1: Replace f(x) with y in the original function.
y = 2x^2 - 1
Step 2: Swap the x and y variables.
x = 2y^2 - 1
Step 3: Solve the equation for y.
Rearrange the equation to isolate y:
2y^2 = x + 1
Divide both sides by 2:
y^2 = (x + 1)/2
Take the square root of both sides, considering both positive and negative solutions:
y = ±√((x + 1)/2)
Therefore, the equation of the inverse function is:
f^(-1)(x) = ±√((x + 1)/2)