SOLUTION: How do I solve a problem like this? {{{f(x)=(x - 1)^2 + 3}}} a. Find the domain and range of f b. Find {{{f^(-1)}}} c. Find the domain and range of {{{f^(-1)}}} d. Is {{{f^(

Algebra ->  Functions -> SOLUTION: How do I solve a problem like this? {{{f(x)=(x - 1)^2 + 3}}} a. Find the domain and range of f b. Find {{{f^(-1)}}} c. Find the domain and range of {{{f^(-1)}}} d. Is {{{f^(      Log On


   

Question 690993: How do I solve a problem like this?
f%28x%29=%28x+-+1%29%5E2+%2B+3
a. Find the domain and range of f
b. Find f%5E%28-1%29
c. Find the domain and range of f%5E%28-1%29
d. Is f%5E%28-1%29 a function?
Answer by mouk(232) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=%28x+-+1%29%5E2+%2B+3
a) Find the domain and range of f
domain of f is set of values of x that can be used. There are no values of x for which the function misbehaves, so domain is -infinity%3C=x%3C=infinity
range of f is the set of values that the function f produces.
Now, %28x+-+1%29%5E2+%3E=0+ for all values of x
so, %28x+-+1%29%5E2+%2B3%3E=3+for all values of x
hence, range is f%28x%29%3E=3 or 3%3C=f%28x%29%3C=infinity

b) Find f%5E%28-1%29
Let y=%28x+-+1%29%5E2+%2B+3, we need to re-arrange and make x the subject of the equation

y=%28x+-+1%29%5E2+%2B+3
%28x+-+1%29%5E2+=+y-3
x-1+=+%22%22+%2B-+sqrt%28+y-3+%29+
x+=+1+%2B-+sqrt%28+y-3+%29+
This is the inverse function, so f%5E%28-1%29%28x%29+=+1+%2B-+sqrt%28+x-3+%29+

c. Find the domain and range of f%5E%28-1%29
domain of f%5E%28-1%29 is set of values of x that can be used. So x-3%3E=0
so domain is x%3E=3
range of f%5E%28-1%29 is the set of values that the function f%5E%28-1%29 produces, this is all real values, so -infinity%3C=f%5E%28-1%29%3C=infinity
Also, note that the domain of f = range of f%5E%28-1%29
and, range of f = domain of f%5E%28-1%29
so finding are consistent with (a)

d. Is f%5E%28-1%29 a function?
A function must provide a 1:1 mapping, so because of the plus/minus sign then f%5E%28-1%29 will produce two values for any given input value of x.
Hence, f%5E%28-1%29 is not a function.