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

If the function 𝑓 : 𝑓(𝑥) = 5𝑥 − 3, then the variation function 𝑉(ℎ) = ＿ at 𝑥 = 2.

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Video Transcript

If the function 𝑓 is such that 𝑓 of 𝑥 is equal to five 𝑥 minus three, then the variation function 𝑉 of ℎ is equal to what at 𝑥 is equal to two.

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

Next, if we substitute 𝑥 is equal to two into our function 𝑓 of 𝑥, 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 ℎ is seven plus five ℎ minus seven. And since seven minus seven is zero, that’s equal to five ℎ. The variation function 𝑉 of ℎ for the function 𝑓 of 𝑥 is five 𝑥 minus three at 𝑥 is equal to two is therefore 𝑉 of ℎ is equal to five ℎ.