In mathematics, a function is a relation between a set of inputs (domain) and a set of possible outputs (range) such that each input is related to exactly one output. When graphed, functions typically appear as curves or lines on a coordinate plane, with the independent variable (usually denoted as x) plotted along the horizontal axis and the dependent variable (usually denoted as y) plotted along the vertical axis.

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## Characteristics of a Function Graph

### 1. One-to-One Correspondence

**A function must exhibit a one-to-one correspondence between its inputs and outputs.** This means that for every value of x, there should be only one corresponding value of y. In other words, each input has a unique output. If a graph fails to satisfy this condition, it is not considered a function.

### 2. Vertical Line Test

**The vertical line test is a method used to determine whether a graph represents a function.** If any vertical line passes through the graph at more than one point, then the relation is not a function. On the other hand, if every vertical line intersects the graph at most once, then the relation is indeed a function.

### 3. Continuous Behavior

**Functions exhibit continuous behavior on their graphs.** This means that there are no breaks, jumps, or holes in the graph. A continuous function can be drawn without lifting the pen from the paper. Discontinuities, such as asymptotes or jumps, indicate non-function behavior.

## Different Types of Function Graphs

### 1. Linear Functions

**A linear function is a function whose graph is a straight line.** The general form of a linear function is y = mx + b, where m is the slope of the line and b is the y-intercept. The graph of a linear function is a straight line that extends infinitely in both directions. Linear functions have a constant rate of change.

### 2. Quadratic Functions

**A quadratic function is a function that can be represented by a parabolic graph.** The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola that opens either upwards or downwards. Quadratic functions can have one or two x-intercepts.

### 3. Exponential Functions

**An exponential function is a function where the variable is in the exponent.** The general form of an exponential function is y = a*b^x, where a and b are constants. The graph of an exponential function is characterized by exponential growth or decay. Exponential functions have a horizontal asymptote.

### 4. Trigonometric Functions

**Trigonometric functions are functions involving trigonometric ratios such as sine, cosine, and tangent.** The graphs of trigonometric functions are periodic, meaning they repeat their values at regular intervals. Trigonometric functions have specific amplitude, period, and phase shift properties.

## Identifying the Function Graph

When given a set of graphs, it is important to determine which graph represents y as a function of x. The following characteristics can help in identifying the function graph:

### 1. Vertical Line Test

**Apply the vertical line test to each graph.** If a vertical line intersects the graph at more than one point, then the graph does not represent a function. If every vertical line intersects the graph at most once, then the graph represents a function.

### 2. One-to-One Correspondence

**Check if there is a one-to-one correspondence between the inputs and outputs.** For each input value of x, there should be only one corresponding output value of y. If multiple outputs are associated with the same input, then the graph does not represent a function.

### 3. Continuous Behavior

**Examine the graph for any breaks, jumps, or holes.** Functions have smooth, continuous behavior on their graphs. Discontinuities indicate non-function behavior.

## Practice Problems

Let’s solve some practice problems to identify which graphs represent y as a function of x:

### Problem 1

Given the following graphs, determine which one represents y as a function of x:

Analysis:

- Graph 1 passes the vertical line test and shows one-to-one correspondence. It represents a function.
- Graph 2 fails the vertical line test as a vertical line intersects it at two points. It does not represent a function.
- Graph 3 passes the vertical line test but doesn’t show one-to-one correspondence. It does not represent a function.

### Problem 2

Determine whether the following graphs represent y as a function of x:

Analysis:

- Graph 4 fails the vertical line test and does not have one-to-one correspondence. It does not represent a function.
- Graph 5 passes the vertical line test and shows one-to-one correspondence. It represents a function.
- Graph 6 passes the vertical line test but has breaks in its graph. It does not represent a function due to discontinuities.

## Conclusion

Identifying which graph represents y as a function of x is crucial in understanding the relationship between variables. By applying the vertical line test, checking for one-to-one correspondence, and examining the continuous behavior of the graph, you can determine whether a graph represents a function or not.

Remember that functions have specific characteristics that set them apart from non-functions. Practice solving problems and analyzing graphs to enhance your understanding of functions and their graphical representations.