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

Question

$f (x) = {\begin{matrix} 5 & if 0 < x \leq 2 \\ 20 - 3 [[2 - 4 x]] & if x > 2 \end{matrix}, for x = 5.6$

Accepted Answer

Hey, everyone in this problem, we're asked to evaluate the value of the following function where X is equal to 6.4. The function we're given is a piecewise function. We have FMX is equal to eight if zero is less than X, which is less than, or equal to four. And we have F of X is equal to 15 minus three, multiplied by the greatest integer of four minus eight, X if X is greater than four. And the problem tells us that these double square brackets indicate the greatest integer function. And sometimes that function is also called the floor function. So if you've heard that in your course, those are interchangeable. Now we have four answer choices. Option A is 158. Option B is 159. Option C is eight and option D is negative 129. Now, the X value we were given X is equal to 6.4 and we have a piecewise function. So we need to figure out which piece of this function we're using X is equal to 6.4 which is greater than four. And so our piecewise function tells us that if X is greater than four, we use this second piece. And so the function that we're using is F FX is equal to 15 minus three, multiplied by the floor or greatest integer of four minus eight X. And we're just gonna substitute in our X value. OK. So our function evaluated at that particular X value is equal to 15 minus three, multiplied by the flow of four minus eight, multiplied by 6.4. Now, before we can evaluate our function, we need to evaluate this flower function. Our greatest integer function before we can do that, we need to simplify inside of those brackets. So we have that our function evaluated at the particular point we're looking at is equal to 15 minus three multiplied by the floor. We have four minus eight multiplied by 6.4. If we simplify this, we get negative 47.2 and now we can take the floor or the greatest integer of that value. Now, if we think of it as a floor function, OK? The floor is beneath your feet. So you wanna find an integer that's beneath or below the value that you have. OK. We have negative 47.2 an integer below that is negative 48. OK? If you want to think of it as the greatest integer function, OK. Let's draw a number line and we're gonna put negative 49 negative 48 negative 47 negative 46 on our number line from left to right and Warner draw our value of negative 47.2. So that's gonna be between negative 47 and negative 48. A little closer to negative 47. Now, the greatest integer function tells us to find the greatest integer less than the value you're at. OK. So we're on the number line and we want a number that's less than that means we want to move to the left and less than numbers are on the left. So we're gonna take our value negative 47.2 and we're gonna move to the left until we hit our first integer. And the integer we hit is gonna be negative 48. OK? So no matter which way you think about this function, you're still gonna get that same value. And in this case, it's negative 48. And so F evaluated at 6.4 is gonna be equal to 15 minus three, multiplied by negative 48. This is equal to 15 plus 144. And we were subtracting a negative value. So that turns into a plus and we get that, this is equal to 159. So the function F of X evaluated at X equals 6.4 gives us a value of 159 which corresponds with answer choice B. 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.
$f (x) = {\begin{matrix} 5 & if 0 < x \leq 2 \\ 20 - 3 [[2 - 4 x]] & if x > 2 \end{matrix}, for x = 5.6$

Key Concepts

Function Notation and Evaluation

Piecewise Functions

Domain and Conditions in Functions

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