Input Output

'e' in an equation does not mean that is the variable you are looking for, you must read the example carefully to determine which variable you are solving for. 'e' & 'x' are numbers in Algebra

* how to solve for 'e' & 'x' in your TI calculator: * Click on the "2nd" button > then click on the "Ln" button (this is for 'e' to 'x' * Then enter the exponents if applicable

Example: the function V(m) models the number of visitors to a theme park (in thousands) during month 'm' of the year (m= 1 corresponds to January, 2 to February and so on). Interpret the equation V(6)=8

*Step 1: note positioning. 'V' is OUTPUT & 'm' is INPUT *Step 2: note the number '6' is in the input place for 'm' which equals the month > if 1 corresponds with month January, then the 6 would correspond with JUNE since that is the 6th month *Step 3: note the number '8' is in the OUTPUT place & the output is 'V' which represents the number of visitors *Step 4: 6 = input & month of JUNE 8 = output & 8,000 visitors to a theme park * Answer: There were 8,000 visitors in June

Example: Gina is bicycling as part of a fitness routine. The number of minutes, 'm', that she spends cycling determines the number of calories, 'C', that she burns. Gina knows that a 50 minute ride will burn 450 calories. How can we express this in function notation

*Step 1: note the formula since this is a word problem instead of the equation. F(X)= output support *Step 2: note the positioning of the numerical value. > 'm' (number of minutes) = is the Cause & the INPUT > 'C' (number of calories burned) = is the Effect & the OUTPUT *Step 3: plug in the values. * Answer: C(50) = 450

Positioning F(a)=b

- Output (input) = supports output - Example: C(State)= Capital C(New Mexico)= Santa Fe [is correct] C(Austin) = Texas [is wrong]

(Input, output)

- the left is always going to be the input & the right is always the output. - Example: (Denver, Colorado) is correct. (Colorado, Denver) would be wrong

How to interpret/solve formula or information on a Graph. **Horizontal Axis (X-axis): will always represent your INPUT VARIABLE (which is the variable inside the parenthesis) **Vertical Axis (Y-axis): will always represent your OUTPUT VARIABLE Example: the function M(t) models the number of members of a club, 'M', in thousands, 't' years after 2010. Evaluate M(7) and Interpret

1. First note that the number given is inside the parenthesis therefore, this is an INPUT value and would be on the X AXIS 2. Next, on the graph, go to the X-Axis > look for the input value number, in this case it's 7 3. Next, draw a line up to the line that's created on the graph 4. Next, on the Y-Axis (output) > look on which number this landed on, on yhe Y-axis 5. The X-axis 6. Then interpret the answer based on the positioning of the input & output value in the problem. 7. In this case, the M was outside the parenthesis which will be the output variable 8. The 't' was inside the parenthesis so this is the INPUT variable 9. In this problem, the output is 9 & the input is 7. Next, interpret the answer: the club has 9,000 members in 2017

*Tricky example - note: always look at algebra questions as military time and not regular AM & PM time* The number of customers, 'N', in a store 't' hours after midnight is given by the function N(t) = -t² + 28t - 171 How many customers will be in the store at 4pm?

1. Look at the time as military time (24 hour clock, which starts at midnight) > then count, 4 PM on a 24 hour clock is 16:00. So, 't' is 16 2. Plug in 16 for 't' 3. Equation will look like: N(16) = -16² + 28(16) - 171 4. Plug in the equation in the calculator after the '=' which will be: -16² + 28(16) - 171. (Which will be 21) 5. Answer: N(16) = 21

Jimmy opened a lemonade stand and he noticed that the number of sales, S is influenced by the temperature, 't', according to the function: S(t) = 0.6t + 5 How many sales will Jimmy make on a 70 degree day?

1. Plug in '70' in the formula in place of 't' (why? Because 't' is for temperature and the example mentions 70 degree day) 2. The formula will look like this-> S(70) = 0.6(70) + 5 3. Next, calculate the equation after the '='. Which will look like -> 0.6 * 70 + 5 = 47 4. Answer: S(70) = 47 5. Note, you do not multiply the variable you are solving for

How to solve equation with a numerator & denominator (as a division format)

1. Plug in the value for the variable 2. Then solve from right to left in segments (make sure to solve exponents separately as a segment) 3. Solve from bottom to top (denominator to numerator) 4. Then once you solve both the top & bottom > then divide the numerator by the denominator 5. Then the result is your answer

The input and output are usually numerical, but they don't have to be. See example C(State) = Capital

C(New Mexico) = Santa Fe Why? Because the capital of New Mexico is Santa Fe and we were solving for the Capital

multivariate situation: Brant sells only two products in his computer shop: laptops and tablets. Laptops (denoted as L) cost $400 each, while tablets (denoted as T) cost $250 apiece. In Brant's case, the total revenue is the output he is concerned about, while the sale of laptops and tablets are inputs. Since you are looking at three total variables here (R, L, and T), this is a multivariate situation. In fact, you can represent the total revenue, R, with the following function:

R(L,T)=400L + 250T R(5,3)=(400×5)+(250×3), R(5,3)=2000+750, R(5,3)=2,750.

F(x)

Read as "F of X" - F is the OUTPUT in this case - X is the INPUT in this case (input,output)

Example: Hans is going hiking today. The function E(t) models his elevation, 'E', in thousands of feet, after 't' hours of hiking. What does the ordered pair (8, 11) tell us.

Step 1: note the positioning for the function E(t): 'E' is Output & 't' is Input Step 2: note the positioning for the ordered pair (8,11): '8' is Input & '11' is Output Step 3: plug in the ordered pair to the function based on the positioning. > E = elevation in thousands of feet & this is OUTPUT > 11 is output = 11 thousand of feet > t = hours of hiking & this is INPUT > 8 is Inout = 8 hours of hiking * Answer: Hans will be 11,000 feet high after 8 hours of hiking

For multivariate situations, interpret function notation in a similar fashion to two-variable situations. For example, f(x,y) means that f is the one dependent variable for the two independent variables x and y.

True

INPUT is the Cause OUTPUT is the Effect

True

Multivariate

involving more than two variables

Example: F(a) = b

note that the result of the equation will go hand in hand with the Output variable