Question

I need help, I was gone from school due to illness.

Use the function; y = 3x - 2 for questions 1, 2 and 3.

1) Write the function in function notation using the name h.

2) Evaluate the function for x = -5. Express your answer in function notation.

3) For what value of x does h(x) = 4?

4) Consider the sequence 9, 12, 15, 18 … . What is f(7)?

Answer 1

1. h(x) = 3x -2
2. h(-5) = -17
3. x=2
4. 27

Explanation:

"Function notation" works like this. A function is described with a name and a list of arguments. The arguments are the independent variables.

<function name>(<argument>, <argument>, ...)

Often, we use letters from the middle of the alphabet for function names, like f, g, h, j. Sometimes, a letter is used that suggests what the function computes: p(x) to mean a polynomial in x; a(x) to indicate area as a function of x, or the x-term of an arithmetic sequence.

To evaluate a function for a particular argument or set of arguments, simply replace the variables in the function definition with the corresponding argument value:

f(x, y) = 10x +y . . . . . . . . . . example function with two arguments

f(3, 2) = 10(3) +2 = 32 . . . . position in the list tells the corresponding variable

__

When a function is graphed, the ordered pairs used for the graph are ...

(x, f(x))

That is, the function value is graphed on the vertical axis, and the independent variable used for the function argument is graphed on the horizontal axis.

__

1)

The function name is h, the function argument name is x, and the definition is given by the equation y=3x-2. The "functional form" of the equation will be ...

h(x) = 3x -2

__

2)

h(-5) = 3(-5) -2 = -15 -2

h(-5) = -17

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3)

Substitute 4 for h(x), and solve for x.

h(x) = 3x -2

4 = 3x -2 . . . . h(x) = 4

6 = 3x . . . . . . add 2

2 = x . . . . . . . divide by 3

Then we have ...

h(2) = 4 . . . . . . . . the value of x is 2 for h(x) = 4

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4)

We notice the first term of the sequence is 9, and the common difference is 12-9 = 3. The general term of an arithmetic sequence is ...

an = a1 +d(n -1)

In functional form, this will be ...

f(x) = 9 +3(x -1)

f(x) = 3x +6

Then f(7) is ...

f(7) = 3(7) +6 = 21 +6

f(7) = 27

We can also find f(7) by extending the sequence 3 more terms:

9, 12, 15, 18, 21, 24, 27

_____

Additional comment

When you're working with variables and functions, there can be a tendency to get confused by the parentheses. For example, x(y) can mean "x times y" or it can mean "the value of function x when y is its argument". So, it is important to understand when a particular token is a variable or a function name. The context is generally helpful. When f(x) appears by itself on the left side of an equal sign, it is generally signifying that what is on the right is the definition of a function:

f(x) = <some expression involving x>

In that context, it is certainly not indicating the product of (f) and (x).

Also, it is helpful to have an understanding that function names generally come from the middle of the alphabet (f, g, h, j), while variable names generally come from the end of the alphabet (w, x, y, z). There are always exceptions.

Function names can be used by themselves in certain contexts. For example, the sum of functions might be written ...

`(f+g)(x)\equiv f(x)+g(x)`

The function of a function f(g(x)) is called a "composition". That can be indicated by the circle operator:

`(f\circ g)(x) \equiv f(g(x))`

This is different from the product of two functions:

`(f\cdot g)(x)\equiv f(x)* g(x)`