Question

Let f be a linear function such that f(1) = 5 and f(3) = 9. Find an equation for f(x).

Answer 1

Answer:

$f(x)=2x+3$

Step-by-step explanation:

So we know that:

$f(1)=5 \text{ and } f(3)=9$

In other words, we have 2 coordinates: (1,5) and (3,9).

With this, we can figure out the slope. Let (1,5) be x₁ and y₁ and let (3,9) be x₂ and y₂. The slope formula is:

$m=\frac{y_2-y_1}{x_2-x_1}$

Substitute:

$m=\frac{9-5}{3-1}\\m=4/2=2$

So, the slope is 2.

Now, use the point-slope form to figure out the y-intercept. The point-slope form is:

$y-y_1=m(x-x_1)$

For consistency, let's continue to use (1,5) as x₁ and y₁. Now, we can use 2 as m. Thus:

$y-(5)=2(x-1)$

Distribute:

$y-5=2x-2$

Add 5 to both sides:

$y=2x+3$

So:

$f(x)=2x+3$

And that's our equation :)