Question

A function \(y=f(x)\) is increasing on the interval \([0,5] .\) What conclusions can you draw about the graph of \(y=f^{-1}(x) ?\)

Short answer

The graph of \(y = f^{-1}(x)\) is increasing on the interval corresponding to \(f([0, 5])\).

Step-by-step solution

- Understand the Function Behavior

The function \(y = f(x)\) is given to be increasing on the interval \[0, 5\]. This means that for any \(x_1 < x_2\) in this interval, we have \(f(x_1) < f(x_2)\). An increasing function never decreases in value over its domain.

- Analyze the Inverse Function Relationship

Recall that the graph of the inverse function \(y = f^{-1}(x)\) is a reflection of the graph of \(y = f(x)\) across the line \(y = x\). This means that if \(y = f(x)\) is increasing, then \(y = f^{-1}(x)\) will also be increasing because the ordering of pairs is preserved.

- Characterize the Graph of the Inverse Function

Given that \(f(x)\) is increasing on \[0, 5\], we conclude that \(f^{-1}(x)\) is also increasing on the described interval, but mapped on the set of \(y\)-values of \(f(x)\) in \[0, 5\]. This means every \(y_1 < y_2\) ensures \(f^{-1}(y_1) < f^{-1}(y_2)\).

Key concepts

Increasing Function

When we say a function is increasing on an interval, it means that as the input (or x value) of the function increases, the output (or y value) also increases. For our function given in the exercise, this implies that for every pair of points within the interval \[0, 5\], if we pick any two points where one x value is smaller than the other (\(x_1 < x_2\)), their corresponding y values will satisfy \(f(x_1) < f(x_2)\).

An increasing function never decreases over its specified interval. This is key when considering the properties of its inverse function. An important point to remember is that an increasing function will not have any plateau (flat) or decreasing segments within the given interval. Hence, its graph constantly rises from left to right.

Graph Reflection

One fascinating aspect of inverse functions is how their graphs relate. The graph of an inverse function \(y = f^{-1}(x)\) is a mirror image of the original function \(y = f(x)\) when reflected over the line \(y = x\). In essence, the original function swaps its x and y coordinates when we look at the inverse function.

For example, if \(f(a) = b\), then the inverse function will satisfy \(f^{-1}(b) = a\). This swapping of coordinates can be visualized as a reflection over the line \(y = x\).

Since our original function \(f(x)\) is increasing, its graph rising from left to right ensures that the reflected graph \(f^{-1}(x)\) will also rise from left to right once reflected over \(y = x\). This reflects the preservation of the order of pairs and means the ordering of points in the inverse is maintained, implying the inverse function is also increasing.

Interval Analysis

To understand how the inverse function behaves over an interval, let’s revisit the original interval where \(f(x)\) is increasing. For our function \(f(x)\), this interval is \[0, 5\]. This means we are considering the function’s behavior specifically within this range of x values.

Now, when we analyze the inverse function \(f^{-1}(x)\) for the same interval, we look at the corresponding y values obtained from the original interval \[0, 5\]. Since \(f(x)\) is increasing on \[0, 5\], its y values will also be ordered, implying that \(f^{-1}(x)\) will map increasing y values back to x values.

Therefore, every time \(y_1 < y_2\) in the original function’s output within \[0, 5\], we will have \(f^{-1}(y_1) < f^{-1}(y_2)\). This analysis confirms that both the original function and its inverse are increasing over their respective intervals. Understanding how these intervals translate between original and inverse functions is crucial for graphing and problem solving.