Can you help me with this problem

Find the inverses of each of the functions below algebraically.


a.p(r)=2r2+2r−1

b.3y+5x=18

p = 2r^2+2r-1

swap variables
r = 2p^2+2p-1
2p^2+2p-(1+r) = 0
p = [-2 +/- sqrt(4+8(1+r))]/4
= [-2 +/- sqrt(8r+12)]/4
= [-1 +/- sqrt(2r+3)]/2

So, what's with the 2 values? The original function fails the horizontal-line test, so it has no single inverse. However, if the domain is restricted to a single branch of the parabola (the vertex is at r = -1/2) then the inverse is one of the two values calculated.

Of course! I can help you find the inverses of these functions step-by-step.

a. For the function p(r) = 2r^2 + 2r - 1:

Step 1: Replace p(r) with y.
y = 2r^2 + 2r - 1

Step 2: Swap the variables r and y.
r = 2y^2 + 2y - 1

Step 3: Solve for y.
To solve for y in terms of r, we need to rearrange the equation:
2y^2 + 2y - 1 = r

Step 4: Quadratic formula.
Using the quadratic formula, we have:
y = (-2 ± sqrt(2^2 - 4(2)(-1 - r))) / (2(2))
Simplifying this expression gives:
y = (-1 ± sqrt(1 + 2r)) / 2

Therefore, the inverse function is p^(-1)(r) = (-1 ± sqrt(1 + 2r)) / 2.

b. For the equation 3y + 5x = 18:

Step 1: Replace y with x and x with y.
3x + 5y = 18

Step 2: Solve for y.
To isolate y, we need to move the variables to one side:
5y = 18 - 3x

Step 3: Divide both sides by 5.
y = (18 - 3x) / 5

Therefore, the inverse function is x = (18 - 3y) / 5.

I hope this helps! Let me know if you have any further questions.

Of course! I'd be happy to help you find the inverses of the given functions.

a. To find the inverse of the function p(r) = 2r^2 + 2r - 1, we need to swap the roles of r and p(r) and solve for r.

Step 1: Replace p(r) with y: y = 2r^2 + 2r - 1.

Step 2: Swap the roles of y and r: r = 2y^2 + 2y - 1.

Step 3: Solve for y. To do this, we can rearrange the equation to isolate y:

2y^2 + 2y - 1 - r = 0.

Step 4: We can solve this quadratic equation using the quadratic formula:

y = (-b ± √(b^2 - 4ac))/(2a).

Applying this formula to our equation, where a = 2, b = 2, and c = -(r + 1), we have:

y = (-(2) ± √((2)^2 - 4(2)(-(r + 1)))) / (2(2)).

Simplifying further:

y = (-2 ± √(4 + 16r + 8)) / 4.

y = (-2 ± √(8r + 12)) / 4.

Step 5: Simplify the expression by factoring out a 4 from the numerator:

y = -1/2 ± √(2r + 3) / 2.

Finally, we have obtained the inverse of the function p(r), which is given by:

p^(-1)(r) = -1/2 ± √(2r + 3) / 2.

b. To find the inverse of the function 3y + 5x = 18, we'll first solve for y in terms of x and then swap the roles of x and y.

Step 1: Start by isolating y on one side of the equation:

3y = 18 - 5x.

Step 2: Divide both sides by 3:

y = (18 - 5x) / 3.

Step 3: Now, swap the roles of x and y:

x = (18 - 5y) / 3.

The inverse of the function 3y + 5x = 18 is given by:

x = (18 - 5y) / 3.