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TExES Mathematics 7-12 (235): Practice & Study Guide48 chapters | 435 lessons | 42 flashcard sets

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Lesson Transcript

Instructor:
*Anna Teper-Dillard*

I was born and raised in Poland. Before I ended up settling in Chicago, I traveled through Europe and Asia. I have been to China, Mongolia, Russia, Italy, Spain, Germany, Czech Republic and England. In my free time, I love reading, and being physically active. I ski, bike, swim and run long distance. I finished the Chicago Marathon, the Fox Valley Marathon, a couple of Half-Marathons and many shorter races. In the summer of 2017, I will be participating in 70.3 miles Iron Man triathlon in Michigan. Currently, I live in Woodrigde, IL with my husband and three children. I have two master degrees, one in Secondary Education and the other in Science of Mathematics. My philosophy on teaching and learning :) I am a long-life learner. Discovering new knowledge (in any subject) is fascinating and rewards us with a broader and more informative point of view. Being exposed to wider range of experiences gives us an edge in life. Although grades we earn throughout our education do not communicate the whole story about who we are, they are good indicators of many attributes and skills we have accomplished: perseverance, fortitude, hard, consistent work, collaboration, reflection on our own mistakes and yes, humbleness.

In this lesson, you will learn what domain and range are and how to find them in absolute value and polynomial functions with inequalities. You will look at domain and range expressed as intervals, inequalities, and shaded graphs.

**Domain** is the set of all *x* values, the independent quantity, for which the function *f(x)* exists or is defined. For example, if we take the linear function:

*f(x)* = 2*x* + 3, we can evaluate *f(x*) at any point, and we will get a real answer for *y*:

Thus, the domain of *f(x)* is all real numbers, or negative infinity to infinity.

**Range** is the set of all *y* values, the dependent quantity, that will result from substituting all *x* values (the domain) into the function.

So the range of *f(x)* = 2*x* + 3 is also all real numbers, because no matter what value of *x* is, we can always multiply that number by 2 and add 3.

That was an equation, but how will an inequality change the domain and range of this linear function? Well, we'll see that the inequality does not affect the domain and range of linear functions at all. Let's not confuse domain and range with a solution to an inequality. These two concepts are different. **Domain and range** mean all possible values of *x* and *y* that could be substituted to the inequality. A **solution** means all possible values that make the inequality statement true.

Let's look at an example:

*f(x)* > 2*x* + 3

The domain is still all real numbers, because the inequality sign changes the relationship between the *y* and the *x*, and not the actual set of values we could substitute for either variable. The solution to the inequality states that the *y* values have to be greater than *2x + 3*, not equal. So while the range is still all real numbers, the solution is the shaded area, for which *y* will always be greater than the line of the function. Let's check a couple of examples:

As we see, whatever *y* value we picked, there will always be an *x* value that will make the inequality true. Therefore, domain and range of linear inequalities will always be all real numbers, which again, is not the same as the solution to the inequality.

**Absolute value** is the distance from zero, regardless of the direction. Therefore, the absolute value is always positive. The absolute value function uses the absolute value symbol (two parallel lines) to express only positive output for either positive or negative input.

The domain is all real numbers, because an absolute value is still a linear function. The range, however, will depend on the **vertex** of the absolute value function (the minimum or the maximum). The vertex of an absolute value function (and quadratic as well) is the lowest or the highest possible *y* value. With this particular inequality, the function has a minimum, and the range is all *y* values greater than the minimum.

Let's look at another example:

A **polynomial function** is a combination of coefficients, variables and exponents, where the exponent has to be a positive integer (a whole number). For example, the basic quadratic function is a polynomial *f(x)* = *x*2.

The domain of this function is all real numbers, because you can substitute any number for the independent quantity *x*. However, the range of the quadratic function is restricted, because if you square any number, the resulting value will always be positive, as shown in this table. Therefore, the range of this particular quadratic function starts at 0 (the vertex, the minimum of the function) and goes to positive infinity.

How will it change as an inequality? Let's take a look!

As we can see on the graph, the *y* values are greater than the parabola (graph of a quadratic function), but what about the *x* values? Remember that you can read inequality from two sides: *y* is greater than the *x* values means that the *x* values have to be less than the *y*. To algebraically approach finding domain and range, we need to follow several steps. Let's look at them now:

Step 1. Set the inequality to 0 instead of *y* or *f(x)*:

Step 2. Factor the polynomial completely, then set each factor equal to zero to find the x-intercepts (zeros) of the function.

Step 3. Sketch a number line and plot the zeros of the function (found from factoring). Depending on the number of zeros, you'll have a different number of intervals divided by the zeros. In this case, we have three intervals:

Step 4. Determine the sign of the function *f(x)* in the leftmost subinterval, by picking any *x* value within the subinterval, substituting it into the equation, and then determining if the *y* is positive or negative. For example:

*x* values from negative infinity to 2

Step 5. Repeat step 3 for all intervals between the zeros of the function:

Second Interval: *x* values from 2 to 9

Third Interval: *x* values from 9 to infinity

Step 6. Write the domain based on the intervals with true statements.

Range of the polynomial function of an even degree (quadratic *y* = *x*2, quartic *y* = *x*4 or any even power) will always depend on the minimum or maximum value of *y*.

Then follow the next steps of testing the signs in each of the possible intervals. Plot the zeros on the number line:

There are three intervals, so test points from each one using the inequality statement:

The domain and range of a linear inequality is always all real numbers, regardless of the sign of inequality. The range of absolute value inequalities will depend on the vertex.

With **polynomial inequalities**, intervals of the *x* values created from zeros will determine the domain and range. Follow the 6 steps from this lesson to determine them.

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TExES Mathematics 7-12 (235): Practice & Study Guide48 chapters | 435 lessons | 42 flashcard sets

- What is an Inequality? 7:09
- Finding the Domain & Range of Functions with Inequalities 13:08
- Graphing Inequalities: Practice Problems 12:06
- What is an Absolute Value? 4:42
- How to Evaluate Absolute Value Expressions 5:28
- How to Solve an Absolute Value Equation 5:26
- Solving Absolute Value Practice Problems 7:09
- Graphing Absolute Value Functions
- How to Graph an Absolute Value and Do Transformations 8:14
- Graphing Absolute Value Equations: Dilations & Reflections 6:08
- How to Solve and Graph an Absolute Value Inequality 8:02
- Solving and Graphing Absolute Value Inequalities: Practice Problems 9:06
- Go to Inequalities and Absolute Values

- Go to Triangles

- Go to Circles

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