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Understanding Functions: What Makes a Relation a Function?

Explore the concept of functions, their properties, and how to evaluate functions using functional notation. Learn how to distinguish functions from relations and apply the vertical line test.

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Understanding Functions: What Makes a Relation a Function?

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  1. Functions f(x)=2x-7 g(x)=x+12

  2. A function is a special kind of relation. (A relation is an operation, or series of operations, that maps one number onto another.)

  3. A function is a special kind of relation. Functions are the relations that mathematicians study most. What makes a relation a function? Good question!

  4. In a function, each input value can give only one output value. Does that make sense? Then try this:

  5. If you put a number into a function, you want to know that only one number can come out.

  6. Consider the relation that subtracts 11 from a number:

  7. If you put a number in, is there only one possible number that could come out?

  8. and have more than one outcome? For any input value, there is only one possible output value. Is there any number you can subtract 11 from So this is a function.

  9. The most common way to write this function would be,

  10. When we write it like this, x is always the input value, and y is always the output value,

  11. If we wanted to find out what y is when x is 42, we would substitute 42 for x, and then calculate y.

  12. So we could say that 42 belongs to the domain of the function, and 31 belongs to the range.

  13. The domain of a function is the set of all input, or x-, values. The range of a function is the set of all possible output, or y-, values.

  14. One common way to write a function uses function notation. Function notation looks like this: Read that, “F of x equals two x plus 1.”

  15. Given Now it becomes obvious that x is the input variable. You may be asked questions such as, find f (-2).

  16. It’s just a matter of substitution. Given If you replace every x with –2, find f(-2).

  17. It’s just a matter of substitution. Given If you replace every x with –2, you can easily calculate:

  18. It’s just a matter of substitution. If you replace every x with –2, you can easily calculate: Given

  19. Given find:

  20. Functions

  21. Functions A function is a relation in which no two ordered pairs have the same first coordinate.For every x there is only one y. (1, 2) (2, 4) (3, 6) (4, 8) A relation that is a FUNCTION (1, 2) (2, 4) (2, 5) (3, 6) A RELATION that is not a function 2 3 4 1 2 3 1 2 3 2 3 4 2 3 4 1 2 3 RELATION FUNCTION FUNCTION

  22. Functions Vertical Line Test:If no two points on a graph can be joined by a vertical line, the graph is a function. Function Relation Function

  23. Functional Notation An equation that is a function may be expressed using functional notation. The notation f(x) (read “fat (x)”) represents the variable y. E.g., y = 2x + 6 can be written as f(x) = 2x + 6. Given the equation y = 2x + 6, evaluate when x = 3. y = 2(3) + 6 y = 12 For the function f(x) = 2x + 6, the notation f(3) means that the variable x is replaced with the value of 3. f(x) = 2x + 6 f(3) = 2(3) + 6 f(3) = 12

  24. Evaluating a Function Given f(x) = 4x + 8, find each: 1.f(2) 2.f(a) f(a) = 4(a) + 8 = 4a + 8 f(2) = 4(2) + 8 = 16 3.f(a + 1) 4.f(-4a) f(-4a) = 4(-4a) + 8 = -16a+ 8 f(a + 1) = 4(a + 1) + 8 = 4a + 4 + 8 = 4a + 12

  25. Evaluating a Function If f(x) = 3x - 1 and g(x) = 5x + 3, find each: 1. f(2) + g(3) 2. f(4) - g(-2) = [3(2) -1] + [5(3) + 3] = 6 - 1 + 15 + 3 = 23 = [3(4) - 1] - [5(-2) + 3] = 11 - (-7) = 18 3. 3f(1) + 2g(2) = 3[3(1) - 1] + 2[5(2) + 3] = 6 + 26 = 32

  26. Evaluating a Function If g(x) = 2x2 + x - 3, find each: 1. g(2) g(2) = 2(2)2 + 2 - 3 = 8 + 2 - 3 = 7 2. g(x + 1) g(x + 1) = 2(x + 1)2 + (x + 1) - 3 = 2(x2 + 2x + 1) + x + 1 - 3 = 2x2 + 4x + 2 + x - 2 = 2x2 + 5x

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