The police can determine the speed a car was traveling from the length of the skid marks it leaves. The function they use is S = f(L) = 2sqrt(5L) where S is speed (mph) and L is the length of the skid marks (feet)
(a) If skid marks of length 125 feed are measured, what was the speed of the car
Inputting 125 in the formula i get 50 mph
(b) find the formula for the inverse function.
f^-1(S) = ?
How would i find the inverse?
a) is correct
b) since the S and L are defined variables, taken the inverse in the usual way changes their definitions.
We could just follow the usual steps ...
if S = 2√(5L) , the inverse is obtained by interchanging the variables, so
inverse is
L = 2√(5S)
we would then solve this new equation for S
L^2 = 4(5S)
L^2 = 20 S
S = L^2/20 , which is totally meaningless and false for our defined values of S and L, as speed and length of skid mark.
I think b) is poorly stated and they probably meant to solve for L in terms of S
which would be:
S = 2√(5L)
S^2 = 4(5L) = 20L
L = S^2/20
notice if we plug in 50 for the S as speed
we get L = 50^2/20 = 125, which was out initial input
82.697
To find the inverse of a function, you need to interchange the roles of the variables. In this case, you have the function S = f(L) = 2√(5L), where S is the speed (mph) and L is the length of the skid marks (feet).
To find the inverse, follow these steps:
1. Start with the original function:
S = 2√(5L)
2. Swap the variables S and L:
L = 2√(5S)
3. Solve for S to get the inverse function:
Square both sides to eliminate the square root:
L^2 = 20S
Divide both sides by 20:
S = L^2/20
Therefore, the inverse function is:
f^-1(S) = L^2/20, where S is the speed (mph) and L is the length of the skid marks (feet).
To find the inverse of a function, follow these steps:
1. Replace the function notation with y.
Now the function becomes y = 2√(5L).
2. Swap the x and y variables.
Instead of y = 2√(5L), write x = 2√(5y).
3. Solve the equation for y.
Start by squaring both sides to get rid of the square root:
x^2 = 4(5y).
Divide both sides by 4:
x^2/4 = 5y.
Divide both sides by 5:
y = x^2/20.
4. Replace y with f^(-1)(x).
The inverse function is f^(-1)(x) = x^2/20.
So, the formula for the inverse function is f^(-1)(S) = S^2/20.