If <em>x</em> = -1, you have
2(-1) + 3 cos(-1) + <em>e</em> ⁻¹ ≈ -0.0112136 < 0
and if <em>x</em> = 0, you have
2(0) + 3 cos(0) + <em>e</em> ⁰ = 4 > 0
The function <em>f(x)</em> = 2<em>x</em> + 3 cos(<em>x</em>) + <em>eˣ</em> is continuous over the real numbers, so the intermediate value theorem applies, and it says that there is some -1 < <em>c</em> < 0 such that <em>f(c)</em> = 0.
the assumption being that the first machine is the one on the left-hand-side and the second is the one on the right-hand-side.
the input goes to the 1st machine and the output of that goes to the 2nd machine.
a)
if she uses and input of 6 on the 2nd one, the result will be 6² - 6 = 30, if we feed that to the 1st one the result will be √( 30 - 5) = √25 = 5, so, simply having the machines swap places will work to get a final output of 5.
b)
clearly we can never get an output of -5 from a square root, however we can from the quadratic one, the 2nd machine/equation.
let's check something, we need a -5 on the 2nd, so

so if we use a "1" as the output on the first machine, we should be able to find out what input we need, let's do that.

so if we use an input of 6 on the first machine, we should be able to get a -5 as final output from the 2nd machine.

Given that, the length of the garden is 2(x+6)feet and width is 3.5x feet.
So, length : l = 2(x+6) and width = 3.5x.
To find the fence of the garden we need to find the perimeter of the rectangle.
Formula to find the perimeter is:
p = 2l + 2b
= 2*2(x+6) + 2*3.5x
= 4(x+6) + 7x By multiplication.
= 4x + 24 + 7x By distribution property.
= 11x + 24 Combining the like terms.
So, he need to buy 11x + 24 feet fencing.
Hope this helps you!