Function Notation and Systems as Models (4.4) Keeping different lines straight

POD Rewrite the equation into slope-intercept form (y = mx + b): -15x + 3y = 21 What is the slope? The y-intercept? The x-intercept? Graph it.

POD Rewrite the equation into slope-intercept form (y = mx + b): -15x + 3y = 21 3y = 15x + 21 y = 5x + 7 Slope = 5 y-intercept = 7 x-intercept = -7/5

What is function notation? We have written equations using y and x: y = 5x + 7. Function notation simply involves rewriting the y into something like f(x). This means that the x values are being put into the function called f. We say “f(x) is a function of x.” Or, “f of x.)

When it’s useful Suppose we have two linear equations to graph. If we rewrite them into slope-intercept form, we can graph them easily.

When it’s useful But now we have two equations with x and y. That could be confusing– when we’re given an x and have to find a y value, which y and x do we mean? And if we have even more equations, we could get lost. Function notation helps keep it straight.

Graphing When we use function notation, we can still think in terms of x and y. When we graph it, we can change the label on the vertical axis from y to f(x).

When it’s useful Rewrite the equations using function notation. Let’s use the letters f and g.

When it’s useful Rewrite the equations using function notation. Let’s use the letters f and g.

Using function notation It’s actually a little easier to use function notation to find values. Say f(x) = 3x 2. If we want to know what it equals when x = 4, we write f(4) = 3(4) 2 = 48. We can add another function, g(x) = 4x + 1, and find what that equals when x = 4 by writing g(4) = 4(4) + 1 = 17. We don’t need to wonder which equation we’re dealing with because the function notation tells us.

Let’s use it Let f(x) = 8x – 7, and g(x) = 2x 2. Find the following: f(5)f(5) + g(-2) g(5)(g(5))(g(-2) f(-2)f(5)/g(5) g(-2)f(g(-2))

Let’s use it Let f(x) = 8x – 7, and g(x) = 2x 2. Find the following: f(5)= 33f(5) + g(-2) = 41 g(5) = 50(g(5))(g(-2) = 400 f(-2) = -23f(5)/g(5) = 33/50 g(-2) = 8f(g(-2)) = 57

Application Let’s use a system of equations to solve a real world problem. I’ve actually done this. You need to replace your air conditioning unit in the house. After researching your options, you have two choices. One costs $1800 and will cost around $60/month to operate. The other costs $2600, but should cost only $50/month to operate. Write an equation for the cost of each unit. Use function notation.

Application Each unit could have its own equation as a model. f(x) = x g(x) = x When are their costs equal? What is that cost? You could do this using algebra or graphing. Let’s do both.

Application To find out when the costs are equal, set the equations equal to each other x = x 10x = 800 x = 80 So, at 80 months, you would have spent the same on either unit. At 80 months, that cost would be $6600. We can get that number from either equation.