Question Video: Determining the Variation Function of an Exponential Function Mathematics

Video Transcript

Determine the variation function 𝑉 of ℎ of the function 𝑓 of 𝑥 is equal to 𝑒 to the power of three 𝑥 at 𝑥 is equal to two.

In this question, we’re asked to determine the variation function 𝑉 of ℎ of the given function 𝑓 of 𝑥 is equal to 𝑒 to the power of three 𝑥 at a value of 𝑥 is equal to two. And to do this, let’s start by recalling what we mean by the variation function of a given function at a given value. A variation function for a function with 𝑥-inputs measures how much the function changes when its 𝑥-inputs change from 𝑎 to 𝑎 plus ℎ. In general, the variation function 𝑉 of ℎ for a function 𝑓 of 𝑥 at 𝑥 is equal to 𝑎 is 𝑓 evaluated at 𝑎 plus ℎ minus 𝑓 evaluated at 𝑎.

In our case, our function 𝑓 is given as 𝑒 to the power of three 𝑥 and our value of 𝑎 is equal to two. So to determine the variation function 𝑉 of ℎ, we need to substitute 𝑓 of 𝑥 is 𝑒 to the power of three 𝑥 and 𝑥 is equal to two into this equation. And start by substituting 𝑎 is equal to two, we get that 𝑉 of ℎ is equal to 𝑓 evaluated at two plus ℎ minus 𝑓 evaluated at two. And remember, our function 𝑓 of 𝑥 is 𝑒 to the power of three 𝑥. So we need to evaluate this at two plus ℎ. And then we need to subtract this evaluated at two. This then gives us 𝑒 to the power of three times two plus ℎ minus 𝑒 to the power of three times two.

We could stop here. This is a valid expression for 𝑉 of ℎ. However, we can simplify this by evaluating the expressions in the exponents. In the first term, we can distribute three over the parentheses. And in the second term, we know three times two is equal to six. And since three multiplied by two plus ℎ is six plus three ℎ, we get 𝑒 to the power of six plus three ℎ minus 𝑒 to the sixth power.

And once again, we could stop here. However, we can simplify this further by noting both of these two terms share a factor of 𝑒 to the sixth power. This is because 𝑒 to the power of 𝑛 multiplied by 𝑒 to the power of 𝑚 is equal to 𝑒 to the power of 𝑛 plus 𝑚. So we can rewrite the first term as 𝑒 to the sixth power multiplied by 𝑒 to the power of three ℎ. We can then take out the shared factor of 𝑒 to the sixth power in the first and second terms. This gives us 𝑒 to the sixth power multiplied by 𝑒 to the power of three ℎ minus one. And we can’t simplify this any further. This then gives us our final answer.

The variation function 𝑉 of ℎ of the function 𝑓 of 𝑥 is equal to 𝑒 to the power of three 𝑥 at 𝑥 is equal to two is 𝑉 of ℎ is equal to 𝑒 to the sixth power multiplied by 𝑒 to the power of three ℎ minus one.