1.1. The function, f(x) is defined by f(x) = $\frac{1}{{e}^{x}}$-x+1 $x\inℝ$

Show that the equation f(x) = 0 can be written in the form

x = e^-x + 1

1.2. On the same diagram sketch the graphs of y = x and y = e^-x + 1.

1.3. The equation f(x) = 0 has a root, a, close to x = 1.
The iterative formula x_n+1 = ${e}^{-{x}_{n}}$+1 with x_o = 2 is to be used to find ⍺ correct to three significant figures

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root ⍺.

1.4. (i) Find the values of x₁, x₂ and x₃, giving each correct to three significant figures.

(ii) How many iterations are required before x_n, and x_n-1, agree to two decimal places?

1.5. The root ⍺ lies in the interval p < x < q.
Write down the values of p and q such that a can be deduced accurate to two decimal places from the interval.