**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*) = *x*^{2} 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*) = *x*^{2} 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*) = 3*x* 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*) = 3*x* has a domain of all real numbers. If any real number can be substituted for *x*, 3*x* 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*) = 5*x* + 2 and *g*(*x*) = –2*x*, 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 *m*^{2} + *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 *m*^{2} + *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, *m*^{2} + *n* = (2)^{2} + (–2) = 4 – 2 = 2.

If *m*#*n* is equivalent to *m*^{2} + 2*n*, 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* = *n*^{2} + 2*m*. 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* + 3*n*, 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* + 3*n*. That expression, *m* + 3*n*, is now the term after the # symbol: *n*#(*m* + 3*n*). 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* + 3*n*): *n* + 3(*m* + 3*n*) = *n* + 3*m* + 9*n* = 3*m* + 10*n*.

**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* = 2*x* + 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* = –3*x* + 5, the *m* is –3, and the *b* is 5.

### Ask a Question

Have questions about this article or topic? Ask### Related Questions

See More Questions### Popular Articles

- Kindergarten Sight Words List
- First Grade Sight Words List
- 10 Fun Activities for Children with Autism
- Signs Your Child Might Have Asperger's Syndrome
- Definitions of Social Studies
- A Teacher's Guide to Differentiating Instruction
- Curriculum Definition
- Theories of Learning
- What Makes a School Effective?
- Child Development Theories