The graph of f (x)₌√x undergoes a vertical stretch by a factor of k, k›1.The vertically stretched graph would look the same if it had been horizontally compressed by a certain factor. Determine this factor

To determine the factor by which the graph needs to be horizontally compressed, we can use the concept of inverse functions.

When a function is stretched vertically by a factor of k, the y-values are multiplied by k while the x-values remain the same. In this case, the original function f(x) = √x is stretched vertically by a factor of k. So, the new function can be written as g(x) = k * √x.

To find the factor by which the graph needs to be horizontally compressed, we need to find the inverse of the function g(x). The inverse of g(x) can be found by swapping the x and y variables and solving for y.

Let's start by swapping the variables:
x = k * √y

Next, we isolate the square root:
√y = x / k

Squaring both sides:
y = (x / k)^2

So, the inverse function of g(x) is h(x) = (x / k)^2.

Now, we compare h(x) = (x / k)^2 with the original function f(x) = √x. The two functions should have the same shape in order for the vertically stretched graph to look the same as the horizontally compressed graph.

By comparing the two functions, we can see that h(x) = (x / k)^2 is obtained from f(x) = √x by compressing the graph horizontally by a factor of k.

Therefore, the factor of horizontal compression is k.