Question

Let f be a linear function. When f x( ) is divided by x − 3, the remainder is 5. When f x( ) is divided by x − 4, the remainder is 3. What is the value of f ( ) 0 ?

Answer 1

Answer:

f(0)=11

Step-by-step explanation:

f(x) is a linear function, hence it is of the form f(x)=ax+b for some constants a,b.

When we divide f(x) by x-3 the remainder is 5. Moreover, using the division algorith we have that

$f(x)=a(x-3)+(b+3a)$

hence 5=b+3a.

On the other hand, when we divide f(x) by x-4 the reminder is 3. Also, using the division algorithm, we have that

$f(x)=a(x-4)+(b+4a)$

hence 3=b+4a.

Therefore, to find f(x), we have to solve the linear sistem

$b+3a=5\\\\b+4a=3$

If we substract the first and the second equation, we get

$3a-4a=5-3 \quad \Rightarrow \quad -a = 2 \quad \Rightarrow \quad a=-2$

and using the last result we get that

$5=b+3a=b+3(-2)=b-6 \quad \Rightarrow \quad b=11$

Hence, f(x)=-2x+11, and so f(0)=11.