The absolute value is always positive, so you can think of it as the distance from 0. I like to then make the expression on the right hand side without the variables both positive and negative and split the equation that way. In this case our answer is all real numbers, since an absolute value is always positive.
But what we're now going to explore is functions that are defined piece by piece over different intervals and functions like this you'll sometimes view them as a piecewise, or these types of function definitions they might be called a piecewise function definition.
Let's take a look at this graph right over here.
This graph, you can see that the function is constant over this interval, 4x. And then it jumps up in this interval for x, and then it jumps back down for this interval for x. Let's think about how we would write this using our function notation.
Then, let's see, our function f x is going to be equal to, there's three different intervals. So let me give myself some space for the three different intervals. Now this first interval is from, not including -9, and I have this open circle here.
Not a closed in circle. So not including -9 but x being greater than -9 and all the way up to and including I could write that as -9 is less than x, less than or equal to That's this interval, and what is the value of the function over this interval?
Well we see, the value of the function is It's a constant -9 over that interval. It's a little confusing because the value of the function is actually also the value of the lower bound on this interval right over here.
It's very important to look at this says, -9 is less than x, not less than or equal. If it was less than or equal, then the function would have been defined at x equals -9, but it's not. We have an open circle right over there. But now let's look at the next interval. The next interval is from -5 is less than x, which is less than or equal to Over that interval, the function is equal to, the function is a constant 6.
It jumps up here. Sometimes people call this a step function, it steps up.Put in any value more than 3 into 3x-9 and you get a positive.
⎩ ⎨ ⎧ − + − + + = 3 9 2 3 9 2 () x x f x ⎭ ⎬ ⎫ ≥. This means the beginning of the equation is: y = a|x – 0| + 3. y = a|x| + 3. What is a? "The first letter of the alphabet" is true, but unhelpful in this case.
To find the value of a, we can plug in a point from the function into our equation so far and solve. The point (1, 5) will do nicely.
5 = a|1| + 3. a = 2. Tada! The equation for the graph is y = 2|x| + 3.
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This solver (calculator) will try to solve a system. Absolute value equations have two solutions. Plug in known values to determine which solution is correct, then rewrite the equation without absolute value brackets. Plug in known values to determine.
Algebra Examples. Step-by-Step Examples. Algebra. Absolute Value Expressions and Equations. Write the Absolute Value as Piecewise.
More Practice: Use the Mathway widget below to try write an. Piecewise Function. Click on Submit (the blue arrow to the right of the problem) and click on Write the Absolute Value as Piecewise to see the answer. In this section we will give an in depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations. Writing Absolute Value as Piecewise Piecewise: A graph split into multiple sections or written as multiple parts. f(x) = |x| 𝑓𝑥= 𝑥,𝑥≤0.
The piecewise polynomials are when and when. Remove parentheses. Solve for when. Online homework and grading tools for instructors and students that reinforce student learning through practice and instant feedback.