Problem 31 - Precalculus with Limits

Question

In Exercises 23-34, show that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. \(f(x) = 9 - x^2\), \(x \geq 0\) \(g(x) = \sqrt{9-x}\), \(x \leq 9\)

Step-by-Step Solution

Answer

It's demonstrated algebraically since both \(f(g(x))\) and \(g(f(x))\) equal \(x\). Graphically, as the plotted functions are mirror images across line \(y = x\), they are inverse functions.

Step 1: Calculate Composite Functions

Firstly calculate \(f(g(x))\) and \(g(f(x))\).\nFor \(f(g(x))\): Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = 9 - (\sqrt{9-x})^2\.\nSolve this to get \(f(g(x)) = x\.\nFor \(g(f(x))\): Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x)) = \sqrt{9-(9-x^2)}\.\nSolve this to get \(g(f(x)) = x\.

Step 2: Graphical Representation

In the coordinate plane, draw graphs of \(f(x)\) and \(g(x)\).\n\(f(x)\) is a downward-opening parabola originating at (0, 9) and \(g(x)\) is a square-root function starting at (9, 0) and going downwards.\nWe can see that they are reflections of each other across the line \(y = x\), which indicates that they are indeed inverse functions.

Key Concepts

Composite Functions

Composite functions are formed when the output of one function becomes the input for another function. This might sound complicated, but it's like putting one function inside another. For example, given two functions, say, \( f \) and \( g \), the composite function \( f(g(x)) \) means we apply \( g \) first and then apply \( f \) on the result. To demonstrate that two functions are inverses of each other algebraically, we calculate both \( f(g(x)) \) and \( g(f(x)) \). If each of these composite operations returns the original input \( x \) for all values in their domains, we have inverse functions.

In our case, \( f(x) = 9 - x^2 \) and \( g(x) = \( \sqrt{9-x} \)\). When calculating \( f(g(x)) \) by plugging \( g(x) \) into \( f \) and simplifying, we indeed get back \( x \), which is a crucial step in confirming that \( f \) and \( g \) are inverse functions.

Graphical Representation of Functions

Visualizing functions through their graphs can often provide us with immediate insights. The graphical representation of a function is a picture of what the function does - it shows how the function maps every input to its corresponding output. A graph can highlight important features of a function like intercepts, trends, and even complex behavior like periodicity and discontinuity. When graphing two functions to determine if they are inverses, we should look for symmetry across the line \( y = x \).

For instance, graphing our given functions, \( f(x) \) and \( g(x) \), reveals that one is a downward-opening parabola and the other a square-root graph. By examining these curves, it becomes clear that they are indeed mirror images across the line \( y = x \), reinforcing the conclusion made through the algebraic method that these functions are inverses of each other.

Square-Root Function

A square-root function is of the form \( y = \( \sqrt{x} \) \), and its graph looks like half of a sideways parabola. It typically starts at the origin (0,0) and curves upwards to the right. However, the square-root function in our example, \( g(x) = \( \sqrt{9-x} \) \), shifts and reflects the usual pattern. This particular graph begins at the point (9, 0) instead of the origin and extends downwards and to the left.

Understanding the behavior of basic functions like the square-root function helps us predict how their shifts and reflections will appear visually. This prediction is crucial when we graph such functions to check for inverse relationships, as those will reflect across the diagonal \( y = x \).

Downward-Opening Parabola

Usually, parabolas open upwards, but the function \( f(x) = 9 - x^2 \), mentioned in our exercise, creates a downward-opening parabola. It's shaped like an upside-down 'U.' This happens because the coefficient of \( x^2 \) is negative. The highest point of this parabola is the vertex, which in the case of our function \( f \), is at (0, 9).

A downward-opening parabola is an important graphical feature because it shows us that the function has a maximum output value - in this case, 9. When graphed alongside its inverse, in this case, a square-root function, the two should intersect the vertex of the parabola with the starting point of the square-root function, if one is indeed the inverse of the other. The downward-opening parabola of our function \( f \), combined with \( g \) being a square-root function, provides a clear visual indicator that these functions are inverses due to their reflected symmetry along the line \( y = x \) as mentioned earlier.