# Function problem help

#### Zeal

##### New member
If y=f(x)
How do i get the function use to derive these.

100=f(100)

50=f(70)

#### HallsofIvy

##### Elite Member
Given any finite number of values, there exist an infinite number of functions that take on those values. Here, you have two values and you might remember that "there is a unique straight line passing through two given points". So there exist a unique linear function taking on those two values. Such a function can be written as "f(x)= ax+ b". f(100)= 100a+ b= 100 and f(70)= 70a+ b= 50.

Subtracting the second equation from the first eliminates b and gives 30a= 50 so a= 5/3. Then 100a+ b= 500/3+ b= 100 so b= 100- 500/3= -200/3. f(x)= (5/3)x- 200/3= (5x- 200)/3.

Checking, f(100)= (500- 200)/3= 300/3= 100 and f(70)= (350- 200)/3= 150/3= 50.

• Zeal

#### Zeal

##### New member
Hi hallsoflvy, could you elaborate a bit im lost.
Thank

#### MarkFL

##### Super Moderator
Staff member
Hi hallsoflvy, could you elaborate a bit im lost.
Thank
It would be better if you state which part(s) of the explanation you don't understand. You are given two points on the graph of the function:

(70,50) and (100,100)

As HallsofIvy pointed out, there are an infinite number of functions we could find passing through those two points, but the simplest of these is going to be a linear function ($$n$$ points can be uniquely described by a polynomial of degree $$n-1$$). Using the definition of slope, and the point-slope equation for a line, we then find:

$$\displaystyle f(x)=\frac{100-50}{100-70}(x-70)+50=\frac{5}{3}x-70\cdot\frac{5}{3}+50=\frac{5}{3}x-\frac{200}{3}=\frac{5}{3}(x-40)$$

#### mmm4444bot

##### Super Moderator
Staff member
… could you elaborate …
We could, if you could explain where you're stuck.

If you're not familiar with linear equations (eg: lines, slopes, equation forms), let us know. We can help you find online videos and lesson links. #### Zeal

##### New member
It would be better if you state which part(s) of the explanation you don't understand. You are given two points on the graph of the function:

(70,50) and (100,100)

As HallsofIvy pointed out, there are an infinite number of functions we could find passing through those two points, but the simplest of these is going to be a linear function ($$n$$ points can be uniquely described by a polynomial of degree $$n-1$$). Using the definition of slope, and the point-slope equation for a line, we then find:

$$\displaystyle f(x)=\frac{100-50}{100-70}(x-70)+50=\frac{5}{3}x-70\cdot\frac{5}{3}+50=\frac{5}{3}x-\frac{200}{3}=\frac{5}{3}(x-40)$$

Thanks, this cleared my confusion