HA1-439: Functions Intro Remember, a relation is ANY set of ordered pairs like (3,2), (-2, 4), (4.5, 6) …It is any set of x’s and y’s. A FUNCTION is a special relation. A function is a relation for which each value from the set the first components of the ordered pairs (often x’s) is associated with exactly one value from the set of second components of the ordered pair(often y’s). Remember, if you have a relation with repeated x values, it is not a function.

HA1-439: Functions Intro Function notation and equations look like this: f(x) = 3x + 6, which is the same as y=3x+6 This says in English, the function of x is represented by 3x + 6 When you put an x value into the equation, you get a unique y value that you can then graph or work with.

HA1-439 Using Function Notation Function Notation looks like this: f(x) = 3x+6 f(x) = 3x + 6 is the same thing as y = 3x+6 The f(x), or the function of x means……….. If we put a number in for x, what happens?? What do we get for an answer? When you are given a problem to solve, it looks like this Question: Given f(x)=3x+6, what is f(2) This means substitute 2 in for x and see what you get?

Question: Given f(x)=3x+6, what is f(2) Substitute 2 in for x and see what you get? f(x) = 3 (2) + 6 =>> 6 + 6 = 12 So f(2) = 12 INPUT of 2 gives OUTPUT of 12 This could be written as an ordered pair (2,12) and graphed if desired.

An equation in the form y=mx +b has a graph that is a linear function( data in straight line).  Remember!! x = independent variable = input = domain y = f(x) = dependent variable = output/answer = range   To emphasize the function-like properties of this equation, we replace y with another name, like f, g, or “cost.” y = 4x + 5 = f(x) = 4x + 5 . These are the SAME EQUATION   This is called function notation. ( f(x) is read “f of x.” ) So if we are given a member of the domain ( an INPUT), we can find the matching value of the range ( the OUTPUT)

INPUTS of functions Let’s look at examples where you know the INPUT ( x, domain, etc.) of a function, and you need to find the OUTPUT, or f(x), or y value.

Let’s Practice: Given the linear equation f(x)= 3x -6, What is f(0)? For f(x) = 4x+5, if we are given f(3), this means put 3 in for x and generate an output. f(3) = 4(3) + 5 = 12 + 5 = 17. Let’s Practice: Given the linear equation f(x)= 3x -6, What is f(0)?   What is f(1)? You try: Given the linear equation f(x)= 5x + 3, What is f(3)? What is f(4)? f(0)= 3(0) -6 =>> -6 f(1)= 3(1) -6 =>> -3 f(3)= 5(3) + 3 =>> 18 f(4)= 5(4) + 3 =>> 23

Function Tables If we have a table like this We have INPUT (x’s here) AND OUTPUT, answers, f(x) , Like y values, in this table. So data went in, x values, and you get answers, f(x) values, which is the same thing as y values. We don’t always know the equation, although you can figure them out, but often don’t need to.

Function Table Values example: f(-3) If we have a table like this If you’re asked to find f(-3) this means -3 is our INPUT, what’s our answer? You look in the INPUT , or x row, for -3, and then look in the f(x), OUTPUT row for the answer which is 4. INPUT of -3 gives OUTPUT of 4 So here, f(-3) = 4 This basically says if you put -3 into this function, you get 4 as an answer. Remember, you can actually plot this on a graph too!! ( -3,4)

Function Table Examples Use the table to find f(2) Use the table to find g(4) Use the table to find g(-6) If the input, x, is 2 then the output, f(x) is 0. INPUT of 2 gives OUTPUT of 0 If the input, t, is 4 then the output, g(t) is 2. INPUT of 4 gives OUTPUT of 2 If the input, a, is -6 then the output, g(a) is 1. INPUT of -6 gives OUTPUT of 1

OUTPUTS of functions NOW, Let’s look at examples where you know the OUTPUT ( answer, y, range, etc.) of a function, and you need to find the INPUT, or x, or data used to get answers for the function.

Let’s Practice: Given the linear equation Sometimes we are given outputs(range) and have to figure out the inputs(domain). Inputs(domain) look like f(3), or f(4). But Outputs(range) look like f(x) = 3 or f(x) = 5. f(x)=3 is saying, we know the answer is 3, what was the input to GET THAT answer!! Let’s Practice: Given the linear equation f(x)= 3x -6, What is f(x) if f(x) = 18? This means, if we know the answer is 18, what was the input?  Step 1: 18 = 3x – 6 , now solve for x. Step 2: 18+6 = 3x – 6 + 6 =>> 24 = 3x =>> x = 8 So we know that when OUTPUT is 18, INPUT was 8.

For f(x)= 3x -6 , What is f(x) if f(x) = -36 Step 1: Put -36 in for f(x) -36 = 3x – 6 , now solve for x. Step 2: Solve for x -36+6 = 3x – 6 + 6 -30 = 3x x = -10 So we know that when OUTPUT is -36, INPUT was -10.

You try: Given the linear equation f(x)= 5x + 3 What is f(x) if f(x) = 28 What is f(x) if f(x) = -12 X = 5 OUTPUT is 28, INPUT was 5. -12 = 5x + 3 -12 – 3 = 5x +3 -3 -15 = 5x X = -3 OUTPUT is -12, INPUT was -3.

Function Table Values example: f(x) = 0 If we have a table like this If you’re asked to find the f(x) = 0 you look in the OUTPUT or f(x) row for 0, and then look above in the x row , INPUT row, for the answer which would be 2. So here, when f(x) = 0 , x = 2 This basically says if your answer was 0 with this function, your INPUT was 2 . OUTPUT is 0 means INPUT was 2 Remember, you can actually plot this on a graph too!! ( 2, 0)

Function Tables Sometimes with function tables, we are given the OUTPUT and asked to figure out what the INPUT value was. Look at this example. We are asked to find ALL values, t, such that f(t) = 15. We look where f(t) = 15 And then see what the corresponding t values are, The INPUTS, that give us OUTPUTS or answers of 15. SO, when f(t) = 15 , t = { 3 , 11 } Remember, it is possible for a function to have repeated y values ( outputs), it just CAN’T have repeated x values, or INPUTS When OUTPUT is 15, INPUTs were 3 & 11.

Function Tables Sometimes with function tables, we are given the OUTPUT and asked to figure out what the INPUT value was. Look at this example. We are asked to find ALL values, a, such that h(a) = -5. We look where h(a) = -5, NOT where a = -5 And then see what the corresponding t values are, the INPUTS, that give us OUTPUTS or answers of -5. SO, when h(a) = -5 , a = -1 When OUTPUT is -5, INPUT was -1.

Function Tables Find ALL values, t, such that f(t) = 0. We look where f(t) = 0, NOT where t = 0 there is NO f(t) = 0, No Solution. Find ALL values, x, such that q(x) = 0. We look where q(x) = 0, and we see there are three INPUTS that gave an OUTPUT of 0 , {3,6,12} When OUTPUT is 0, INPUTs were 3, 6, 12.

Analyzing Graphs of Functions When we are looking at graphs, we use the same strategy as above to find either the input ,( domain value) ,or the output ( range value) Let’s Practice This is where x is 3 This is saying, we know the input value x ( domain) is 3, what is the range value ( the y value) if x is 3. Now look up and down to see what the y value of the function is that goes with x=3 So we see that if x=3, we look down and y = -2.So the point (3, -2) is part of this function, a solution.

Analyzing Graphs of Functions When we are looking at graphs, we use the same strategy as above to find either the input ,( domain value) ,or the output ( range value) Let’s Practice This is where x is 1 This is saying, we know the input value x ( domain) is 1, what is the range value ( the y value) if x is 1. Now look up or down to see what the y value of the function is that goes with x=1 So we see that if x=1, we look down and y = -4. We don’t count y=-2 because it is an open circle. So the point (1, -4) is a solution for this function.

Analyzing Graphs of Functions Sometimes we are given the range ( y-value) and we have to find the x value that goes with it. Let’s Practice This is saying, we know the output value y ( range) is -1, what are the domain value(s) ( the x value(s)) if y is -1. This is where y is -1 Now look side to side, because x values are horizontal, to see what x values go with a -1 y value. So we see that if y=-1, we look side to side for x values and find that the function crosses y= -1 at x values of {0,2}. So ( 0, -1) and ( 2, -1) are solution points for this function.

Analyzing Graphs of Functions Sometimes we are given the range ( y-value) and we have to find the x value that goes with it. Let’s Practice This is saying, we know the output value y ( range) is -4, what are the domain value(s) ( the x value(s)) if y is -4. This is where y is -4 Now look side to side, because x values are horizontal, to see what x value(s) go with a -4 y value. So we see that if y=-4, we look side to side for x values and find that the function crosses y=-4 at an x value of -2. So the point ( -2, -4) is a solution for this function.