Using Functions

Question

The function g is defined as:
g:x→ a/x + b

where a and b are constants.

If g(-2)=1 and g(3)=6

Find the values of a and b.

Answer 1

This problem will use a system of equations. Recognize the you have 2 equations and 2 unknowns. So the function g(x)=a/x+b can be set up as follows:

 

First at x=-2 from g(-2)=1

 

1=a/(-2) + b

 

and at x=3 from g(3)=6

 

6=a/(3) + b

 

With your two equations solve for the unknowns I'll use substitution:

 

1=a/(-2) + b          (1)

6=a/(3) + b           (2)

 

Solve equation (1) for b

 

b=1+(a/2)

 

Plug into equation (2)

 

6=a/3 + 1 + a/2

 

Solve for a:

 

6-1=a/3+a/2

5=5a/6

6=a

 

Plug a into equation (2)

6=(6)/3 + b      Solve for b

6/2=b

4=b

 

So final answer is:

a=6

b=4

 

You can verify this by plugging a and b into equation (1) or (2)

 

1=(6)/(-2) + 4

1=1  √

6=(6)/(3) + 4

6=6  √

 

 

 

Answer 2

g(x) → a/x +b

g(-2) = 1   ....   a/(-2) + b  = 1

g(3) = 6   ...     a/3 + b = 6

 

We have two equations with 2 unknowns

To get rid of the fractions

Multiply equation 1 by -2:

Multiply equation 2 by 3

 

a - 2b = -2       -2(equation 1)

a + 3b = 18       3(equation 2)

 

solve for a from the top equation...

a = 2b-2

 

substitute that in place of a in bottom equation

2b-2+3b=18

5b=20

b=4

 

a=2b-2=8-2=6

 

a=6, b=4

 

check:  a/x+b=1 if x=-2       a/x+b=6 if x=3

           6/(-2)+4=1               6/3 +4 =6

            -3+4 =1                     2+4=6

                1=1                          6=6