Reference no: EM13912663
1. Let f and g be the functions whose rules are
f(x) = x + 1/2
and
g(x) 2x -3.
(a) Find f o g and g o f, getting each answer in simplified form.
(b) Find (f o g)(2) and (g o f)(2). In each case show how the numerical answer could be obtained in two different ways: one by an extended arrow diagram, and one by applying the formulas found in part (a).
In Problems 2-6, let f be the function given by
f(x) = √(x+4) - 5.
2. Decide if f is one-to-one. One method is to graph it and cheek the horizontal line test.
Determine the domain and range of f.
Determine the domain and the range of f-1.
Find the graph of the inverse of f by throe methods:
(a) By visually reflecting the graph of f across the diagonal line y = x,
(b) By reflecting the endpoint and the intercepts of graph of f, connecting them with a smooth curve, and (attending the curve to give it the same shape as the graph of f.
(c) By producing an equation for the inverse function and graphing it over the known domain of f-1. The formula over its entire implied domain and the inverse function we seek here are not the same, because the inverse has a smaller domain than the maximal domain of the underlying formula.
3. Find a formula for f-1(x) and state the domain.
4. Find f-1(f(-3)) and f(f -1( -3)).
5. Test the graph of f for symmetry across the line y = x.
6. Compare f-1(-3) and [f(-3)]-1. Are they equal?
7. Let f(x) = 5x - 3/4x - 7, assuming f is one-to-one. Find a formula for f-1 (x).
As a check, find: f(1), f(2), f-1(-2/3), and f-1(7). Do these four calculations support the correctness of your formula for f-1? Why?