MA 109 Student Guide

Example 5.43. Warm-up.

Suppose \(f(x)\) is given in the table below:

Now, let's define a new function \(g(x)=f(2x)\text{.}\) We want to construct the table for \(g(x)\text{.}\) Let's follow the same logic we had for the warm-up in section Section 5.3: to fill in the top row, we need to find \(g(-1)\text{.}\) Based on our definition of \(g(x)\text{,}\) we have that \(g(-1)=f(-2(1))=f(-2)\text{.}\) However, at this point we are stuck, because \(x=-2\) is not in our table for \(f(x)\text{.}\) So, we cannot use the same input values for \(g(x)\text{.}\) Instead, we need to reverse engineer what inputs we can use for \(g(x)\) that will match up to the inputs we have for \(f(x)\text{.}\) For example, we know what \(f(-1)\) is, and \(g\left(-\frac{1}{2}\right)=f(2\left(\frac{1}{2}\right))=f(-1)\text{,}\) that means we can use \(-\frac{1}{2}\) as one of our inputs for \(g(x)\) and we have that \(g\left(-\frac{1}{2}\right)=2\text{.}\) We can follow the same logic to get the rest of the table:

Example 5.49.

Below is the graph of \(h(x)\text{:}\)

We want to figure out which of the following is the graph of \(h(2x)\text{.}\) In each, the gray dotted graph is the orignal graph of \(h(x)\) so you can more easily see how it changed.

Since we are looking for \(h(2x)\text{,}\) we are looking for the graph to squish horizontally (towards the \(y\)-axis) by 2, which is graph D. Graph A stretched vertically, which we learned in Section 5.3 would be \(2h(x)\text{.}\) Graph B compressed vertically, which we learned in Section 5.3 would be \(\frac{1}{2}h(x)\text{.}\) Graph C stretched horizontally, which would be \(f\left(\frac{x}{2}\right)\text{.}\)

Example 5.51.

Suppose the solid blue graph below is the graph of \(f(x)\) and the dotted red graph is the graph of \(g(x)\text{:}\)

We want to write a formula for \(g(x)\) in terms of \(f(x)\text{.}\) In other words, we start with \(f(x)\) and figure out how the graph changed to get to \(g(x)\text{,}\) then write how that changes the formula. Since the question asks for the formula "in terms of \(f(x)\)", that means we will write \(f(x)\) somewhere in our answer. Looking at the graph, we see that \(g(x)\) is the graph of \(f(x)\) stretched horizontally (away from the \(y\)-axis) by \(3\text{.}\) Therefore, that means we are dividing by \(3\) inside of the function \(f(x)\text{.}\) So, our final answer is \(g(x)=f\left(\frac{x}{3}\right)\text{.}\)

Example 5.53.

Suppose \(f(x)=x^2+3x-4\) and \(g(x)\) is the same as \(f(x)\) but stretched horizontally by \(7\text{.}\) Let's write a formula for \(g(x)\text{.}\) Since we have an explicit formula for \(f(x)\) in this case, we can write an explicit formula for \(g(x)\text{,}\) too. Since we are stretching horizontally by \(7\text{,}\) that means we are dividing by \(7\) to the inside of the function. So,

So our final answer is \(g(x)=\left(\frac{x}{7}\right)^2+3\left(\frac{x}{7}\right)-4\text{.}\)

Example 5.55.

Suppose \(f(x)=x^2-x+5\) and \(g(x)=(3x)^2-(3x)+55\text{.}\) Let's figure out what transformations took \(f(x)\) to \(g(x)\text{.}\) Comparing the two formulas, we see that in the formula for \(g(x)\text{,}\) all of the \(x\)'s turned into \((3x)\text{.}\) That means that to get from \(f(x)\) to \(g(x)\text{,}\) we must have multiplied by \(3\) on the inside so that \(g(x)=f\left(3x\right)\text{.}\) That means there was a horizontal squish by \(3\text{.}\)