As a math teacher it is my pleasure to explain the concept of function translation and how it impacts the range of a function.
First, let's start by understanding the given function f(x) = x. This function is a linear function that graphs as a straight line with a slope of 1 passing through the origin (0,0) on the coordinate plane. For every x value, there is a corresponding y value that is exactly the same as x since f(x) = x. Therefore, the range of this function includes all y values that can be produced by the function, which means the range is all real numbers. In interval notation, the range of f(x) = x is written as (-∞, ∞).
Now, let's consider what happens when we translate the graph of this function. A translation means we move the entire graph of the function without changing its shape. In this case, the function is translated 6 units to the right and 2 units up to form a new function.
Translating 6 units to the right affects the x-coordinates of the graph, but it does not affect the "steepness" or direction of the line. In other words, translating the function horizontally does not change the range of the function. Similarly, translating 2 units up affects the y-coordinates of the graph, but since the line has an infinite extent in both the positive and negative y-directions, this vertical shift also does not change the range. After the translation, the function still continues infinitely in both the upward and downward y-directions.
Therefore, after translation, the new function continues to have the same range as the original function, which is the set of all real y values.
So, the statement about the range of both functions that is true is:
"The range is the same for both functions: {y | y is a real number}."
This means that regardless of the translation, the range of the linear function f(x) = x and the translated function will always be the set of all real numbers. The other options provided suggest that the range either includes only non-negative real numbers or that it changes as a result of the translation; these are incorrect because the original function f(x) = x already has an infinite range in both the positive and negative direction, and translation does not alter that.
