# High School: Functions

### Interpreting Functions HSF-IF.A.2

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Students should know that function notation isn't as difficult as they think. In fact, it's downright easy. Much easier than, say, learning French. Nous faire confiance.

All they have to do is isolate an equation for y and then replace it with f(x) (read as "f of x"). That's pretty much it.

The function rule is the equation that represents the unique output values. In other words, it explains what's done to the input value to make it the output value. For instance, a function rule can be, "Multiply the input value by 2 and then subtract three from it." We could write this function rule as f(x) = 2x – 3.

For an equation to be a function, x is the independent variable and the y value (or f(x) value) is the dependent variable. This makes sense, since the output value depends on the input value. D'accord?

The plus side about function notation is that we can write the input and output of a function in one line. Let's say the input of x = 2 makes an output of f(x) = 1. Rather than writing each one separately, we can simply say f(2) = 1. To make sure your students remember this, emphasize that the input is 2 because it's in the parenthesis.

Once students are proficient in using this notation, they can begin to apply it to real-life problems. They should also make sure to use function notation consistently. Practice makes perfect, oui?

For instance, a French real estate agent's weekly earnings can be calculated as the output of €400 plus 6% of his weekly commission. This should be interpreted as the function f(x) = 400 + 0.06x. If the students are asked to find the agent's earnings after his weekly sales are €46,900, they should know that they're being asked to find f(46,900), or f(46,900) = 3,214. Ooh la la!

#### Drills

1. If f(x) = 2x – 12, what is the ordered pair when x = -1?

(-1, -14)

Our input is denoted by x and our output, f(x), is defined by the function rule f(x) = 2x – 12. Since our input is x = -1, we're looking for f(-1) = 2(-1) – 12 = -2 – 12 = -14. If our input is -1 and our output is -14, the ordered pair that correctly represents this is (A).

2. If f(x) = 3x2 + 6, what is the ordered pair when x = 3?

(3, 33)

Our input is denoted by x and our output, f(x), is defined by the function rule f(x) = 3x2 + 6. Since our input is x = 3, we're looking for f(3) = 3(3)2 + 6= -3(9) + 6 = 27 + 6 = 33. If our input is 3 and our output is 33, the ordered pair that correctly represents this is (C).

3. If f(x) = -½x + 4, what is the ordered pair when x = -2?

(-2, 5)

We already know that our input, denoted by x, has to be the first number in the ordered pair. So it's either (B) or (D). When we calculate our output for x = -2, we end up with . If our input is -2 and our output is 5, the ordered pair that correctly represents this is (B).

4. If f(x) = -1.5x – 1, what is the ordered pair when x = -2?

(-2, 2)

We already know that our input, denoted by x, has to be the first number in the ordered pair. So it's either (C) or (D). When we calculate our output for x = -2, we end up with . If our input is -2 and our output is 2, the ordered pair that correctly represents this is (C).

5. If , which is the correct function notation when x = 5?

In function notation, f(x) = y. Substitute in the x's and the y's, and we should be good. Since x = 5 in this case, we know that we'll have f(5) = y. We just need to find y, or f(5). We can do that by using the function rule . Our output is , so our answer is (C).

6. If f(x) = -x, which is the correct function notation when x = 1?

f(1) = -1

In function notation, f(x) = y. Substitute in the x's and the y's, and we should be good. Since x = 1 in this case, we know that we'll have f(1) = y. We just need to find y, or f(1). We can do that by using the function rule f(x) = -x. It doesn't take a genius to know our output is -1, so our answer is (B). While (D) is also true for our function rule (since f(-1) = -(-1) = 1), we're asked for the output when x = 1 and for (D), x = -1.

7. You choose to rent a car. The car rental company charges a flat rate of \$20 plus \$0.22 per mile driven. Which function rule applies to this scenario?

f(m) = 20 + 0.22m

We want the output, f(m) to equal the total cost of renting a car based on the number of miles driven, m. We start off with a constant of \$20 and then a slope of \$0.22 multiplied by the miles driven, our input. Since the two values are added together, the function we're looking for will be in the form: total cost = 20 + 0.22(miles driven). The only one that takes this form is (A).

8. GoLean vitamins are sold by mail order only. They cost \$19.99 per bottle plus \$4 shipping and handling. Which function rule applies to this scenario?

f(b) = 4 + 19.99b

The output, f(b) should equal the total cost of ordering b bottles of GoLean vitamins. Since the \$4 shipping and handling fee doesn't depend on b, it's a constant. Ordering one bottle is \$19.99, but b bottles will cost \$19.99b. Since we're adding the two together, the only answer that makes sense is (D).

9. Which statement best describes the function notation f(2) = 10?

The output is 10 when the input is 2.

Since the input is written in parentheses and the output is written on the other side of the equal sign, we know that f(2) = 10 means the input is 2 and the output is 10. The term f(x) describes the function rule or the output of the input x. Since we know the input is 2, this isn't right. An input of 2 also wouldn't make the output f(10) since f(10) implies that 10 is the input.

10. Which statement best describes the function notation f(x + 1) = 5?

The output is 5 when the input is x + 1.

Since the input is written in parentheses and the output is written on the other side of the equal sign, we know that f(x + 1) = 5 means the input is x + 1 and the output is 5. The term f describes the function itself rather than the output. Since x + 1 is in the parenthesis, we know that (C) can't be right, and 5 is the output, so it can't be (A), either. The only answer we're left with is (D).