Question

Dada a função f(x) = ax + b e sabendo-se que f(3) = 5 e f(-2) = -5 calcule f(½)

Answer 1

`f(1/2)=0`

Explanation:

Given this function:

`f(x)=ax+b`

We need to find the value of:

`f(1/2)`

This is really simple because the problem give us the following data:

`f(3)=5\\\\and\\\\f(-2)=-5`

Using this data we can write a 2x2 system of equations like this:

Let:

`f(3)=5=a(3)+b\\\\\rightarrow3a+b=5\hspace{10}(1)`

and

`f(-2)=5=a(-2)+b\\\\\rightarrow-2a+b=-5\hspace{10}(2)`

Our system will be:

`3a+b=5\hspace{28}(1)\\-2a+b=-5\hspace{10}(2)`

We can solve this system using different methods, however I will use elimination method because I believe it is really simple. So:

`(1)-(2)\\\\5-(-5)=3a-(-2a)+b-b\\\\10=5a\\\\Solving\hspace{3}for\hspace{3}a\\\\a=(10)/(5) =2`

Replacing
`a` into (1)

`3(2)+b=5\\\\6+b=5\\\\Solving\hspace{3}for\hspace{3}b\\\\b=5-6=-1`

Now we found the values of
`a` and
`b`, our function is given by:

`f(x)=2x-1`

Let's verify it:

`f(3)=2(3)-1=6-1=5`

`f(-2)=2(-2)-1=-4-1=-5`

As you can see the function satisfies the data provided by the problem. So we can conclude that it is correct.

Finally, let's find
`f(x)` for
`x=1/2` :

`f(1/2)=2((1)/(2) )-1=(2)/(2) -1=1-1=0`

Hence:

`f(1/2)=0`