Find k

[nycmathguy]

The function f(x) = k(2 − x − x^3)
has an inverse function, and f ^(-1)(3) = −2. Find k.

Note: f^(-1)(3) means f inverse of 3.

Can you set this up for me?

 

[MathLover1]

The function f(x) = k(2 -x - x^3)
has an inverse function, and f ^(-1)(3) = −2. Find k.

Note: f^(-1)(3) means f inverse of 3.

inverse:
y = k(2 -x - x^3)
x= k(2 -y - y^3)
x/k= 2 -y - y^3
y^3+y= 2 -x/k

f ^(-1)(3) = −2

solution:
k=1/4

 

[MathLover1]

One way to think about inverse functions is how they operate. It isn't always possible to algebraically write down an inverse function. Sometimes it is just messy (like the one above). Sometimes it is impossible.

Looking for an x value that gives the output value f(x) is called inverting. It is convenient to introduce a letter to hold the output.
For instance, f(x)=y.
We say the inverse function is the one that takes y-values and gives back x-values. We write that as f^-1(y)=x. For example f(2)=4 we would say f^-1(4)=2. It goes just backwards.

In your problem, there is an unknown k in the function f(x) = k(2 -x - x^3). We're told that
f^-1(3)=-2. In other words, y=3 and x=-2.

so, f(-2)=3

3= k(2 -(-2) - (-2)^3)
3= k(2 +2 +8)
3= k(12)
k=3/12
k=1/4

 

[nycmathguy]

The function f(x) = k(2 -x - x^3)
has an inverse function, and f ^(-1)(3) = −2. Find k.

Note: f^(-1)(3) means f inverse of 3.

inverse:
y = k(2 -x - x^3)
x= k(2 -y - y^3)
x/k= 2 -y - y^3
y^3+y= 2 -x/k

f ^(-1)(3) = −2

solution:
k=1/4

Very impressive reply.

 

[nycmathguy]

One way to think about inverse functions is how they operate. It isn't always possible to algebraically write down an inverse function. Sometimes it is just messy (like the one above). Sometimes it is impossible.

Looking for an x value that gives the output value f(x) is called inverting. It is convenient to introduce a letter to hold the output.
For instance, f(x)=y.
We say the inverse function is the one that takes y-values and gives back x-values. We write that as f^-1(y)=x. For example f(2)=4 we would say f^-1(4)=2. It goes just backwards.

In your problem, there is an unknown k in the function f(x) = k(2 -x - x^3). We're told that
f^-1(3)=-2. In other words, y=3 and x=-2.

so, f(-2)=3

3= k(2 -(-2) - (-2)^3)
3= k(2 +2 +8)
3= k(12)
k=3/12
k=1/4

It's always good to know several ways to find the answer.