Problem 78 - Precalculus : Building Concepts and Connections

Question

Can the graph of a function with range \([4, \infty)\) cross the \(x\) -axis?

Step-by-Step Solution

Answer

No, the graph of a function with range [4, ∞) cannot cross the x-axis.

Step 1: Understand range in a function

Range of a function is the set of all output values (y-values). So for a function with range [4, ∞), all the y-values are equal to or greater than 4.

Step 2: Relate range with x-axis

The x-axis corresponds to y = 0. A function crosses the x-axis at a point if and only if the function's value at that point is 0.

Step 3: Determine whether function can cross x-axis

Since all y-values for this function are more than or equal to 4, there is no point where the function equals 0. So, it never crosses the x-axis.

Key Concepts

Understanding the Range of a Function

The range of a function is one of the basic concepts in understanding how functions behave. It refers to the complete set of possible output values (often called y-values) that a function can produce. For example, if you have a function with a range of \([4, \infty)\), this means that the smallest value that the function can output is 4, and it can go as high as infinity. Consequently, all the y-values for this function are at least 4 or greater.

It's important to visualize the range when you're working with function graphs because it gives us a boundary or limit to the possible outputs. Knowing the range helps in determining what the graph of the function looks like on a coordinate plane, particularly for assessing whether certain points or lines, like the x-axis, can be intersected.

Exploring X-axis Crossings

The concept of an x-axis crossing is straightforward – it’s where the graph of a function meets or intersects the x-axis. On a graph, the x-axis represents all points where the y-value is zero. This means that if a function crosses the x-axis, there is a specific input (x-value) where the output or y-value becomes zero.

However, if we have already established that the function's range is \([4, \infty)\), then none of the y-values can be zero. Thus, for this particular function, an x-axis crossing is impossible. An x-axis crossing would require the function to drop down to y = 0 at some point, which contradicts the given range.

Understanding Y-Values of a Function

Y-values in the context of a function refer to the outputs we get after inserting the x-values (inputs) into the function. These values are crucial because they determine where the graph of the function is located on the coordinate plane.

For a function whose range is defined as \([4, \infty)\), every output (or y-value) is part of this interval, meaning they are always greater than or equal to 4. This prevents any y-value from being below 4, which includes y = 0. Thus, when analyzing functions and their graphs, knowing the possible y-values gives clear indicators of where the graph can and cannot go. It helps in predicting behavior like where the graph could potentially touch or intersect the coordinate axes.