SOLUTION: Given that f(x)=ax+b, f(1)=1 and f ^-1(5)=2, find value of a and of b. Hence, find f ^-1(x) and evaluate f^-1(7) and f ^-1(-5 1/2).

Algebra ->  Finance -> SOLUTION: Given that f(x)=ax+b, f(1)=1 and f ^-1(5)=2, find value of a and of b. Hence, find f ^-1(x) and evaluate f^-1(7) and f ^-1(-5 1/2).      Log On


   

Question 1133690: Given that f(x)=ax+b, f(1)=1 and f ^-1(5)=2, find value of a and of b. Hence, find f ^-1(x) and evaluate f^-1(7) and f ^-1(-5 1/2).
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Given that

f%28x%29=ax%2Bb,

f%281%29=1 and
+f+%5E-1%285%29=2,

find value of a+and of+b.

if f%281%29=1=>f%28x%29=1 when x=1
then 1=a%2A1%2Bb=>1=a%2Bb=>solve for a
1-b=a....eq.1

now first find +f+%5E-1%28x%29
f%28x%29=y
y=ax%2Bb.....swap x and y
x=ay%2Bb......solve for y
x-b=ay
y=%28x-b%29%2Fa.........eq.2

=>+f+%5E-1%28x%29=%28x-b%29%2Fa
if +f+%5E-1%285%29=2, we have
+2=%285-b%29%2Fa
+a=%285-b%29%2F2........eq.3

from eq.1 and eq.3 we have:

1-b=%285-b%29%2F2......solve for b
2%281-b%29=5-b
2-2b=5-b
2-5=2b-b
b=-3

now find a
a=1-%28-3%29....eq.1
a=1%2B3
a=4

so, your equation is: f%28x%29=4x-3

now, go to eq.2, plug in a and b
+f+%5E-1%28x%29=%28x-b%29%2Fa
+f+%5E-1%28x%29=%28x-%28-3%29%29%2F4
+f+%5E-1%28x%29=%28x%2B3%29%2F4
+f+%5E-1%28x%29=%281%2F4%29x%2B3%2F4

and evaluate

f%5E-1%287%29=%281%2F4%297%2B3%2F4=7%2F4%2B3%2F4=10%2F4=5%2F2

and f+%5E-1(-5+1%2F2)=. (-11%2F2)
f%5E-1%28-11%2F2%29=%281%2F4%29%28-11%2F2%29%2B3%2F4=-11%2F8%2B3%2F4=-11%2F8%2B6%2F8=-5%2F8