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Question Video: Finding the Variation Function at a Point for a Given Linear Function Mathematics

If the function 𝑓 : 𝑓(π‘₯) = 5π‘₯ βˆ’ 3, then the variation function 𝑉(β„Ž) = οΌΏ at π‘₯ = 2.

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Video Transcript

If the function 𝑓 is such that 𝑓 of π‘₯ is equal to five π‘₯ minus three, then the variation function 𝑉 of β„Ž is equal to what at π‘₯ is equal to two.

We want to find the variation function 𝑉 of β„Ž for 𝑓 of π‘₯ is five π‘₯ minus three at π‘₯ is equal to two. To do this, we recall that the variation function for a function 𝑓 of π‘₯ at π‘₯ is equal to π‘Ž is defined as 𝑉 of β„Ž is 𝑓 of π‘Ž plus β„Ž minus 𝑓 of π‘Ž, where β„Ž is the change in π‘₯. In our case, our initial point π‘₯ is equal to two, and that’s π‘Ž, so that our variation function 𝑉 of β„Ž is 𝑓 of two plus β„Ž minus 𝑓 of two. Now, substituting first π‘₯ is equal to two plus β„Ž into our function 𝑓, we have 𝑓 of two plus β„Ž is five times two plus β„Ž minus three, that is, 10 plus five β„Ž minus three, which is seven plus five β„Ž.

Next, if we substitute π‘₯ is equal to two into our function 𝑓 of π‘₯, we have 𝑓 of two is five times two minus three, which is 10 minus three, and that’s seven. With these values in our variation function, this gives us 𝑉 of β„Ž is seven plus five β„Ž minus seven. And since seven minus seven is zero, that’s equal to five β„Ž. The variation function 𝑉 of β„Ž for the function 𝑓 of π‘₯ is five π‘₯ minus three at π‘₯ is equal to two is therefore 𝑉 of β„Ž is equal to five β„Ž.

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