Define the function by f(x)=(x-1)^2

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Define the function

f:[1,\infty)\to[0,\infty)

f(x)=(x-1)^2,

and define the function

g:[0,\infty)\to[1,\infty)

g(x)=\sqrt{x}+1.

Answer the following questions. (i) Compute

f\bigl(g(x)\bigr)

and simplify the result. (ii) Compute

g\bigl(f(x)\bigr)

and simplify the result. (iii) Using the results of parts (i) and (ii), decide whether

f

and

g

are inverses of each other. Give full justification referring to the stated domains and ranges. (iv) On the same set of axes, sketch the graphs of

y=f(x)

and

y=g(x)

for the given domains. Mark at least three corresponding points and illustrate their symmetry with respect to the line

y=x

. No scale is required, but key features must be shown clearly.