a.Find f(2) and g(2). b.Find f(7) and g(7). c. Is there any value of x that is in the domains of both f and g such that f(x) and g(x) are not equal?,

Question

Consider the functions f(x)= (x*x)-1/x+1 and g(x)= x-1

Answer 1

f(x) = x² - 1/x + 1 and g(x) = x - 1 so

 

a.

 

f(2) = 2² - 1/2 + 1 = 5 - 1/2 = 9/2 = 4.5

 

g(2) = 2 - 1 = 1

 

b.

 

f(7) = 7² - 1/7 + 1 = 50 - 1/7 = 349/7 = 49.857142...

 

g(7) = 7 - 1 = 6

 

c. Both 2 and 7 are common to both domains and satisfy f(x) ≠ g(x)

 

In fact we can get all such x.

 

All non-zero real numbers are in both domains. Setting f(x) = g(x) and solving gives the answer.

 

x² - 1/x + 1 = x - 1

 

x³ - 1 + x = x² - x

 

x³ - x² + 2x - 1 = 0

 

x ≈ 0.56984

 

So all real numbers except for 0 and that root will solve part c.

 

By the way, the root above has exact value

 

x = 1/3 + (1/6) [ ³√(44 + 12√69) + ³√(44 - 12√69) ]