Fourier Transform - Solving for Impulse Response

[Abide]

Homework Statement

I'm trying to Solve for an impulse response h(t) Given the excitation signal x(t) and the output signal y(t)

x(t) = 4rect(t/2)
y(t) = 10[(1-e^-(t+1))u(t+1) - (1-e^-(t-1))u(t-1)]
h(t) = ?

y(t) = h(t)*x(t) --> '*' meaning convolution!

I am unsure how to take the Fourier Transform of the elements in the output signal. I have posted my attempts below and I would like to know if I am going this correctly or not, Thanks!

Homework Equations

Using the multiplication - convolution duality I know that we need to take the Fourier transform of each element giving us the following...

Y(f) = H(f)X(f)

Which then allows us to solve for H(f) by Y(f)/X(f)

The Attempt at a Solution

First I distributed the unit step functions in y(t) giving...

y(t) = 10[u(t+1)-e^-(t+1)u(t+1) - u(t-1) + e^-(t-1)u(t-1)

Now I take the Fourier transform of each element in y(t)

F(u(t+1)) = 1/(jω+(0²))(e^j2∏f)

F(e^-(t+1)u(t+1)) = 1/(jω+(1²))(e^-j2∏f)

I got this by using the following definition of the Fourier Transform
e^-Atu(t) <---> 1/(jω+A²) for A > 0I was curious as to if anyone could give me some insight on whether I am performing these operations correctly or not. I apologize if I left out any information!

 

[hedipaldi]

Find the Fourier transforms of x(t) and y(t),then use the convolution theorem and inverse transform.

 

[Abide]

Ok, but my question was more about whether I am performing the transform correctly , thank you for your response