If #f(x)=1/2x# and #g(x)# is a linear function, such that #x=−3# and #x=−1# when #f(x)=g(x)#. Write a possible function for #g(x)#?

1 Answer
e52fa787
Jun 2, 2018

#g(x) = f(x) = 1/2x#

This is the only answer for #g(x)#

Explanation:

Intuition

Since #f(x)=g(x)# at two points #x=-3,-1#, we can tell that #g(x)# passes through the two points #(-3,-3/2), (-1,-1/2)#

But if #g(x)# is linear, that means that its graph is a straight line.

And since there is only one way to draw a straight line through two points, there is only one answer for #g(x)#, and since #f(x)# is a straight line passing through those two points as well, #g(x)# has to be equal to #f(x)#

graph{(y-1/2x)=0 [-5, 2, -5, 2]}

More Rigorous Proof

Since #g(x)# is linear, we let #g(x)=ax+b# for constants a & b, then form equations of a & b using the given conditions

Substituting #f(-1)=g(-1)#, we have

#1/2(-1)=a(-1)+b#

#-1/2=-a+b#

Substituting #f(-3)=g(-3)#, we get

#1/2(-3)=a(-3)+b#

#-3/2=-3a+b#

If you solve these two simultaneous equations, you get #a=1/2,b=0# as the only solution

Hence #g(x)# can only be #1/2x#

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