Find the inverse of g(x) = 3x^2 - 5
swap variables and solve for g
x = 3g^2 - 5
x+5 = 3g^2
g = ±√((x+5)/3)
so you have to pick the domain of g which has the branch you want.
Well, to find the inverse of a function, you usually swap the x's and y's and solve for y. But in this case, I'm sorry to inform you, g(x) = 3x^2 - 5 isn't very friendly with inverses. It's kind of like trying to find the inverse of a grumpy cat. Trust me, it's not going to end well. So, instead of finding the inverse, maybe we can introduce this function to some Purr Therapy.
To find the inverse of a function, we need to switch the roles of x and y and solve for the new y.
Step 1: Replace g(x) with y:
y = 3x^2 - 5
Step 2: Swap x and y:
x = 3y^2 - 5
Step 3: Solve for y:
Add 5 to both sides:
x + 5 = 3y^2
Divide both sides by 3:
(x + 5)/3 = y^2
Take the square root of both sides:
√((x + 5)/3) = y
Step 4: Replace y with f^(-1)(x) to denote the inverse function:
f^(-1)(x) = √((x + 5)/3)
So, the inverse of g(x) = 3x^2 - 5 is f^(-1)(x) = √((x + 5)/3).
To find the inverse of a function, you need to switch the roles of x and y and solve for y.
Step 1: Replace g(x) with y
y = 3x^2 - 5
Step 2: Swap x and y
x = 3y^2 - 5
Step 3: Solve for y
Rearrange the equation to isolate y:
x + 5 = 3y^2
Divide both sides by 3:
(x + 5)/3 = y^2
Step 4: Find the square root of both sides
Since we are taking the square root, remember to consider both positive and negative square roots:
√((x + 5)/3) = ± y
Step 5: Swap y and x
y = ± √((x + 5)/3)
Therefore, the inverse of g(x) = 3x^2 - 5 is:
g⁻¹(x) = ± √((x + 5)/3)