Composite and inverse functions

Exercise

Question 1

Consider f(x) = 4-x^{2} , g(x)=\sqrt{x + 3} , h(x) =\frac{1}{2x}. Evaluate the following.
(a) (f \circ g)(1)
(b) (g \circ h)(1)
(c) (f \circ g)(x)
(d) (g \circ h)(x)
(e) (h \circ g)(x)
(f) (f \circ g \circ h)(x)


Question 2

Using the functions given in the previous question, explain why (f \circ g)(-4) does not and cannot exist.


Question 3

Let s(x)=\sqrt{x} and t(x)=x^{2} + 2x + 1. Evaluate (s \circ t)(x) .


Question 4

Show the following function pairs are inverses:
(a) f(x) = x^{2} and g(x) = \sqrt{x}
(b) f(x) = \frac{1}{x^{2}-1} and g(x) =\sqrt{\frac{1}{x} + 1}
(c) f(x) = 2x-2 and g(x) = \frac{x}{2}+1


Question 5

Find the inverses of the following functions.
(a) y = 3x + 2 , (b) y =\frac{1}{4-x}
(c) y = \frac{x+2}{x+5} , (d) y = x^{3}+ 1


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