# 1.2 - Functions and Graphs

## Find function values, or outputs, using a formula or a graph.

### Using a Formula

#### Example Problem:
Consider the linear function $f(x) = 2x + 3$. Find the function value $f(4)$.

#### Solution:
1. **Identify the formula for the function:** $f(x) = 2x + 3$.
2. **Substitute the given input (x-value) into the formula:** To find $f(4)$, substitute $x = 4$ into the formula.
3. **Calculate:** $f(4) = 2(4) + 3 = 8 + 3 = 11$.
4. **Interpret the result:** The function value of $f$ when $x = 4$ is 11. This means that when the input is 4, the output of the function $f$ is 11.

### Using a Graph

#### Example Problem:
Given the graph of a function $f(x)$ below, find $f(2)$.


#### Solution:
1. **Locate the given input on the x-axis:** Find where $x = 2$ on the x-axis.
2. **Find the corresponding point on the function:** From $x = 2$, move vertically until you reach the point where the function $f(x)$ exists.
3. **Determine the y-value at this point:** The y-coordinate of this point is the function value $f(2)=4$.
4. **Interpret the result:** The point you reach on the function is $A=(2, 4)$ graphed below. This means $f(2) = 4$, indicating that when the input is 2, the output of the function $f$ is 4.


### Conclusion

Finding function values can be approached by substituting values into a formula or by reading values off of a graph. Both methods are fundamental skills in mathematics that allow students to understand how functions behave and how to interpret them in various contexts. Practice with both approaches will help students gain confidence in working with functions.

## Graph functions.

## Graphing Functions

Graphing functions is a key skill in mathematics that enables us to visualize the relationship between variables. It involves plotting points on a coordinate plane and connecting them to form a graph that represents the function.

### Steps to Graph a Function

1. **Identify the function equation:** Begin with understanding the function equation you need to graph. This is the mathematical representation of the relationship between the dependent (y) and independent variable (x).

2. **Create a table of values:** Select a range of values for the independent variable (usually `x`) and calculate the corresponding values of the dependent variable (`y`) based on the function equation.

3. **Plot the points:** On a coordinate plane, plot each pair of `(x, y)` values from your table.

4. **Connect the points:** If the function is continuous (like most polynomial functions), connect the points with a smooth curve. For discrete functions, simply plot the individual points.

5. **Label your graph:** Include important features such as the function's equation, intercepts, and any other notable points (e.g., maxima, minima).

### Example Problem: Graph the Function $y = x^2 - 4$

#### 1. Function Equation
The equation we'll graph is $y = x^2 - 4$.

#### 2. Table of Values with Work
Below is a table showing how to calculate the `y` values by substituting the `x` values into the function equation, resulting in specific points on the graph.

| x | Calculation | $y=f(x)$ | Point |
|----|----------------|-----|--------|
| -3 | $(-3)^2 - 4=9-4=5$ | 5 | (-3, 5)|
| -2 | $(-2)^2 - 4=4-4=0$ | 0 | (-2, 0)|
| -1 | $(-1)^2 - 4=1-4=-3$ | -3 | (-1, -3)|
| 0 | $(0)^2 - 4=0-4=-4$ | -4 | (0, -4)|
| 1 | $(1)^2 - 4=1-4=-3$ | -3 | (1, -3)|
| 2 | $(2)^2 - 4=4-4=0$ | 0 | (2, 0) |
| 3 | $(3)^2 - 4=9-4=5$ | 5 | (3, 5) |

#### 3. Plotting Points
Using the "Point" column from the table, plot each point on a coordinate plane.


#### 4. Connecting Points
Since the function $y = x^2 - 4$ is a continuous function that describes a parabola, connect all the plotted points with a smooth curve.


#### 5. Labeling
Make sure to label the graph with the function's equation, $y = x^2 - 4$, and highlight any important points, such as the vertex of the parabola at (0, -4).

### Conclusion

This example demonstrates how to graph the function $y = x^2 - 4$ by calculating and plotting specific points. The process involves creating a table of values, plotting those values on a coordinate plane, and then connecting them to reveal the shape of the function. Through practice, this method can be applied to graph various types of functions, helping to visualize their behavior and properties.

## Determine whether a graph is that of a function by The Vertical Line Test

The vertical line test is a simple yet powerful tool to determine whether a curve in the coordinate plane is the graph of a function. This test is based on the fundamental definition of a function: every input (or x-value) must be associated with exactly one output (or y-value).

### Understanding the Test

1. **Concept:** If a vertical line intersects a curve at more than one point, then the curve does not represent a function. This is because, at that x-value, there are multiple y-values, which violates the definition of a function.

2. **Application:** To use the vertical line test, imagine drawing vertical lines (parallel to the y-axis) at various points along the x-axis. If any of these lines intersect the curve more than once, the curve is not the graph of a function.

### Example

Consider a graph with a curve and we want to determine if it represents a function.

- **Function Curve:** A parabola opening upwards, such as the graph of $y = x^2$, will pass the vertical line test. No matter where you draw a vertical line across the x-axis, it will only ever touch the parabola at one point.

- **Non-Function Curve:** A circle, will not pass the vertical line test. Vertical lines drawn through the circle will intersect it at two points in most locations, indicating that for some x-values, there are multiple y-values.

### Visual Example

- For $y = x^2$, **ANY** vertical line you draw will intersect the curve at exactly one point, confirming it's a function.



Note that one input (x) gives one output (y), for all x.

- For the circle $x^2 + y^2 = 25$, vertical lines through most parts of the circle will intersect at two points, showing it's not a function according to the vertical line test.



Note that we found one input (x=3) gives two outputs (y=4,y=-4).

### Conclusion

The vertical line test is an effective method to visually inspect whether a graph represents a function. It helps in distinguishing functions from other types of relations by ensuring every x-value has a unique y-value. Understanding and applying this test is fundamental in graph analysis and function identification.

## Inequalities

### Given the inequality $x<2$. This represents all real numbers less than two. Here is a table showing the different x values and whether they make the inequality true or false.

|$x$ | $x = 0$ | $x = 1$ | $x = 1.9$ | $x = 2$ | $x = 2.1$ | $x = 3$ | $x = 4$ |
|-------------------|----|------------------|---------------------|-------------------|---------------------|-------------------|-------------------|
| $x < 2$ | $0 < 2$ | $1 < 2$ | $1.9 < 2$ | $2 < 2$ | $2.1 < 2$ | $3 < 2$ | $4<2$
| True or False | True | True | True | False | False | False | False |

This can be represented with the following number line graph:


Since $x=2$ does not satisfy the inequality, it is excluded from the set described by the inequality. That is why $x=2$ has an open circle and not a filled in one.

In interval Notation this can be written as a pair of endpoints in increasing order. Since all x values to the left of 2 are less than 2, the left endpoint is $-\infty$ (all the way infinite far to the left side of the number line). The right endpoint is $x=2$, excluded.

All real numbers less than 2, or $x<2$, can be written in interval notation as the interval $(-\infty,2)$. This signifies all real numbers greater than $-\infty$ and less than $2$.

Similarly, here are the other inequalities and how they look like graphed on a number line.

### $x \leq 2$ or the interval $(-\infty,2]$. This means all real numbers less than or equal to two.

### $x \geq 2$ or the interval $[2,\infty)$. This means all real numbers greater than or equal to two.


### $x > 2$ or the interval $(2,\infty)$. This means all real numbers greater than two.


### Interval Notation Summary Table

| Inequality Type | Inequality Notation | Interval Notation | Verbal Description |
|------------------|----------------------|-------------------|------------------------------------------------|
| Greater Than | $x > a$ | $(a,\infty)$ | All real numbers greater than $a$. |
| Less Than | $x < b$ | $(-\infty,b)$ | All real numbers less than $b$. |
| Greater Equal | $x \geq c$ | $[c,\infty)$ | All real numbers greater than or equal to $c$. |
| Less Equal | $x \leq d$ | $(-\infty,d]$ | All real numbers less than or equal to $d$. |

## Zero Product Property: If $ab=0$, then $a=0$ or $b=0$.

## Find the domain and the range of a function.

### Function 1: $f(x) = x^2 + 2$

#### Domain:
For the function $f(x) = x^2 + 2$, it's a polynomial function. Polynomials are defined for all real numbers. Hence, the domain of this function is $\mathbb{R}$, which represents all real numbers.

### Function 2: $f(x) = \frac{1}{x}$

#### Domain:
The function $f(x) = \frac{1}{x}$ involves division. In division, we cannot divide by zero, as it's undefined. Therefore, any value of $x$ that makes the denominator zero should be excluded from the domain. In this case, $x$ cannot be equal to zero. Hence, the domain of this function is $(-\infty, 0) \cup (0, +\infty)$, which represents all real numbers except zero.

### Function 3: $f(x) = \sqrt{x}$

#### Domain:
The function $f(x) = \sqrt{x}$ represents the square root function. The square root of a negative number is not a real number. Therefore, the input $x$ should be non-negative for the function to be defined in the real number system. Thus, the domain of this function is $[0, +\infty)$, including zero. This means that $x$ can take any value greater than or equal to zero.

### Function 4 $f(x) = \frac{1}{x+4}$:

#### Domain:
For the function $f(x) = \frac{1}{x+4}$, the denominator $x + 4$ cannot equal zero, because division by zero is undefined. Therefore, the only restriction on the domain is that $x + 4 \neq 0$. Solving this inequality for $x$ gives us $x \neq -4$. Thus, the domain of $f(x)$ is all real numbers except $x = -4$, which can be denoted in interval notation as $(-\infty, -4) \cup (-4, \infty)$.

### Function 5 $g(x) = \sqrt{x+5}$:

#### Domain:
For the function $g(x) = \sqrt{x+5}$, the expression under the square root, $x + 5$, must be non-negative, because the square root of a negative number is not a real number. Therefore, the condition $x + 5 \geq 0$ must be satisfied. Solving this inequality for $x$ gives us $x \geq -5$. Thus, the domain of $g(x)$ is all real numbers greater than or equal to $-5$, which can be denoted as $[-5, \infty )$.