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# Functions and Graphs Help (page 2)

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Updated on Oct 27, 2011

### Horizontal Line Test

The horizontal line test can be used to determine how many different x-values, or inputs, return the same f(x)-value. Remember, a function cannot have one input return two or more outputs, but it can have more than one input return the same output. For example, the function f(x) = x2 is a function, because no x-value can return two or more f(x) values, but more than one x-value can return the same f(x)-value. Both x = 2 and x = –2 make f(x) = 4. To find how many values make f(x) = 4, draw a horizontal line through the graph of the function where f(x), or y, = 4.

You can see that the line y = 4 crosses the graph of f(x) = x2 in exactly two places. Therefore, the horizontal line test proves that there are two values for x that make f(x) = 4.

### Domain

All of the x-values of a function, collectively, are called its domain. Sometimes there are x-values that are outside of the domain, but these are the x-values for which the function is not defined.

The function f(x) = 3x has a domain of all real numbers. Any real number can be substituted for x in the equation and the value of the function will be a real number.

The function f(x) = – 4 has a domain of all real numbers excluding 4. If x = 4, the value of the function would be , which is undefined. In a function, the values that make a part of the function undefined are the values that are NOT in the domain of the function.

What is the domain of the function f(x) = √x?

The square root of a negative number is undefined, so the value of x must not be less than 0. Therefore, the domain of the function is x ≥ 0.

### Range

All of the solutions to f(x) are collectively called the range. Any values that f(x) cannot be equal to are said to be outside of the range.

As you just saw, the function f(x) = 3x has a domain of all real numbers. If any real number can be substituted for x, 3x can yield any real number. The range of this function is also all real numbers.

Although the domain of the function f(x) = is all real numbers excluding 4, the range of the function is all real numbers excluding 0, because no value for x can make f(x) = 0.

What is the range of the function f(x) = √x?

You already found the domain of the function to be x ≥ 0. For all values of x greater than or equal to 0, the function will return values greater than or equal to 0.

## Nested Functions and Newly Defined Symbols

### Nested Functions

Given the definitions of two functions, you can find the result of one function (given a value) and place it directly into another function. For example, if f(x) = 5x + 2 and g(x) = –2x, what is f(g(x)) when x = 3?

Begin with the innermost function: Find g(x) when x = 3. In other words, find g(3). Then, substitute the result of that function for x in f(x): g(3) = –2(3) = –6, f(–6) = 5(–6) + 2 = –30 + 2 = –28. Therefore, f(g(x)) = –28 when x = 3.

What is the value of g(f(x)) when x = 3?

Start with the innermost function—this time, it is f(x): f(3) = 5(3) + 2 = 15 + 2 = 17. Now, substitute 17 for x in g(x): g(17) = –2(17) = –34. When x = 3, f(g(x)) = –28 and g(f(x)) = –34.

### Newly Defined Symbols

A symbol can be used to represent one or more operations. A symbol such as # may be given a certain definition, such as "m#n is equivalent to m2 + n." You may be asked to find the value of the function given the values of m and n, or you may be asked to find an expression that represents the function.

If m#n is equivalent to m2 + n, what is the value of m#n when m = 2 and n = –2?

Substitute the values of m and n into the definition of the symbol. The definition of the function states that the term before the # symbol should be squared and added to the term after the # symbol. When m = 2 and n = –2, m2 + n = (2)2 + (–2) = 4 – 2 = 2.

If m#n is equivalent to m2 + 2n, what is the value of n#m?

The definition of the function states that the term before the # symbol should be squared and added to twice the term after the # symbol. Therefore, the value of n#m = n2 + 2m. Watch your variables carefully. The definition of the function is given for m#n, but the question asks for the value of n#m.

If m#n is equivalent to m + 3n, what is the value of n#(m#n)?

Begin with the innermost function, m#n. The definition of the function states that the term before the # symbol should be added to three times the term after the # symbol. Therefore, the value of m#n = m + 3n. That expression, m + 3n, is now the term after the # symbol: n#(m + 3n). Look again at the definition of the function. Add the term before the # symbol to three times the term after the # symbol. Add n to three times (m + 3n): n + 3(m + 3n) = n + 3m + 9n = 3m + 10n.

## Slope and y-Intercept

### Graphing Linear Equations

A linear equation always graphs into a straight line. The variable in a linear equation cannot contain an exponent greater than one. It cannot have a variable in the denominator, and the variables cannot be multiplied.

The graph of a linear equation is a line, which means it goes on forever in both directions. A graph is a picture of all the answers to the equation, so there is an infinite (endless) number of solutions. Every point on that line is a solution.

There are several methods that can be used to graph linear equations; however, you will use the slope-intercept method here.

### Slope of a Line

What does slope mean to you? If you are a skier, you might think of a ski slope. The slope of a line has a similar meaning. The slope of a line is the steepness of a line. What is the y-intercept? Intercept means to cross, so the y-intercept is where the line crosses the y-axis. A positive slope will always rise from left to right. A negative slope will always fall from left to right.

To graph a linear equation, you will first change its equation into slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, also known as y = form. Linear equations have two variables. For example, in the equation y = mx + b, the two variables are x and y. The m represents a number and is the slope of the line and is also a constant. The b represents a number and is the y-intercept. For example, in the equation y = 2x + 3, the number 2 is the m, which is the slope. The 3 is the b, which is the y-intercept. In the equation y = –3x + 5, the m is –3, and the b is 5.

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