Find the value of the function for the given value of x.

Question

$f (x) = {\begin{matrix} 3 & if 0 < x \leq 4 \\ 10 - 2 [[5 - x]] & if x > 4, for \end{matrix}, for x = 6.2$

Accepted Answer

Hey, everyone in this problem, we're asked to evaluate the value of the following function for X is equal to 9.4. The function we're given f of X is a piecewise function. OK. We have FFX is equal to seven If zero is less than X, which is less than, or equal to six. And our function F is equal to 12 minus two, multiplied by the floor of eight minus X if X is greater than six. OK. And in the problem, we're told that the, these double square brackets indicate the greatest integer function. OK, which is also sometimes called the floor function. We're given four answer choices, option a 16, option B seven, option C eight Or option D 14. Now, in this case, we have X is equal to 9.4, an inner function F OK. We decide which piece of this function we used based on the value of X And we need to compare it to six. OK. 9.4 is greater than six. OK. So we have X greater than six, which means that we're gonna be using the second piece of this function. So f of x in this case, is going to be equal to 12 - Multiplied by eight -X and the eight -X is inside of that greatest integer function. Now, we wanna evaluate at XX equals 9.4. So we're gonna substitute X is equal to 9.4. In our equation, we have F evaluated at 9.4 K, the brackets around the 9.4 after our function F indicate that we're evaluating our function F at that particular X value, This is equal to 12 - Multiplied by the floor or the greatest integer of 8 -9.4. Now, before we can evaluate this floor function or the greatest integer function, we need to simplify inside of those brackets. OK? So we have that this is equal to 12 -2 Multiplied by the greatest integer of -1.4. Now, we can think of this two ways if you think of this as a flower function, OK? You hear floor and you think the floor underneath your feet, OK? The floor is beneath you. So the value of the floor function is gonna be the integer beneath the integer we're on Or sorry, not the integer at the value rep. So we're at a value of negative 1.4, the integer below that is going to be negative too. Now, if you think of this as the greatest integer function, we want to take the greatest integer that is less than our value. So if we draw a number line and we think of this, in terms of a number line, we're gonna add negative three, negative two, negative 10 and one to our number line starting from left to right. We're gonna draw the value we have, which is negative 1.4. So it's gonna be between negative one and negative two, a little bit closer to negative one. And again, the greatest integer function wants you to take the greatest integer less than the one that you're at or less than the value that you're at. So we're starting at our value of -1.4. OK? On a number line numbers that are less than are to the left. And so we're gonna move to the left until we hit our first integer. And the first hint integer we hit when we move to the left is negative too. OK? So either way you think of this function, you're gonna get that same result. And in this case, it's negative too. And so our function F evaluated at 9.4 is equal to 12 -2, multiplied by -2. This is equal to 12 plus four. Here we have minus the negative. So that turns into plus four and this is equal to 16. OK? So evaluating our function at X is equal to 9.4 gives us a result of 16, which corresponds with answer choice. A thanks everyone. For watching. I hope this video helped see you in the next one.

Find the value of the function for the given value of x.
$f (x) = {\begin{matrix} 3 & if 0 < x \leq 4 \\ 10 - 2 [[5 - x]] & if x > 4, for \end{matrix}, for x = 6.2$

Key Concepts

Function Notation and Evaluation

Piecewise Functions

Domain and Input Values

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