f is a linear function such that f(1) = 0 and f(2) = 4. How do you write an equation for f(x)
I'm sure you heard somewhere along the line that linear functions can be written as $\displaystyle y=mx+b$ or $\displaystyle f(x)=mx+b$
So we have 2 points, and that's all we need. We use them to plug into that general formula
$\displaystyle 0=m(1)+b$ and $\displaystyle 4=m(2)+b$
Solve those two equations for m and b and you'll be able to write down your line $\displaystyle y=mx+b$
You need to solve the following system of equations
1. 0=m(1)+b and
2. 4=m(2)+b
Now let's try to get you to understand this. I hand you two points, (p,q) and (r,s) and say what's the equation of the line that goes through them
You realize that all lines on the x-y plane can be written as $\displaystyle y=mx+b$
So you also know that for the ordered pair (p,q), the x coordinate is p and the y coordinate is q.... same for (r,s)
So since $\displaystyle y=mx+b$ must be satisfied, you plug your points into in to get
q=mp+b and
s=mr+b
Now in a normal problem p,q,r,s will be numbers so the only "unknowns" here are m and b
You have two equations and 2 unknowns so you can solve for both m and b, and then the line you were looking for is y=mx+b
Here they tell you f(1)=0 and f(2)=4, so since f(x)=y, the points they give you are (1,0) and (2,4)
Hi SHiFT,
If f(1)=0, then the point is (1, 0).
If f(2)=4, then the point is (2, 4)
Find the slope using your two points.
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-0}{2-1}=4$
Now, plug your slope of 4 and point (1, 0) into $\displaystyle y=mx+b$ and find y-intercept b.
$\displaystyle 0=4(1)+b$
$\displaystyle b=-4$
$\displaystyle y=4x-4$
$\displaystyle \boxed{f(x)=4x-4}$
We have 2 conditions, f(1)=0 and f(2)=4
You gave $\displaystyle \boxed{f(x)=\frac{1}{4}x+1}$
And if we compute f(1) we get $\displaystyle \frac{1}{4}(1)+1=1.25\not = 0$
And if we compute f(2) we get $\displaystyle \frac{1}{4}(2)+1=1.5\not=4$
You swapped your x and y's, meaning this function takes 0 to 1 and 4 to 2, you gave us the inverse of the function is question