Question

Show that \(f\) and \(g\) are inverse functions (a analytically and (b) graphically. $$ f(x)=3-4 x, \quad g(x)=(3-x) / 4 $$

Short answer

Algebraic proofs show that \(f(g(x))=x\) and \(g(f(x))=x\) confirming that \(f\) and \(g\) are inverse functions of each other. Graphical representation shows \(g(x)\) as reflection of \(f(x)\) across line \(y=x\).

Step-by-step solution

Algebraic Proof for \(f(g(x))=x\)

Substitute \(g(x)\) into \(f(x)\) to get: \(f(g(x))=f((3-x)/4)=3-4*((3-x)/4). Simplifying this, we get \(x.\)

Algebraic Proof for \(g(f(x))=x\)

Substitute \(f(x)\) into \(g(x)\) to get: \(g(f(x))=g(3-4x)= (3-(3-4x))/4. Simplifying this, we get \(x\). In the algebraic proofs we showed that both \(f(g(x))=x\) and \(g(f(x))=x\) which confirms that \(f\) and \(g\) are inverse functions of each other.

Graphical Illustration

Plot \(f(x)\) and \(g(x)\) on the same graph. Also graph the line \(y=x\). If \(f\) and \(g\) are inverse functions, the graph of \(g\) should be a reflection of the graph of \(f\) about the line \(y=x\). In other words, if you were to fold the graph along the line \(y=x\), the graph of \(f\) and \(g\) should align perfectly, showing that they are indeed inverses of each other.

Key concepts

Algebraic Proof

The concept of an algebraic proof is fundamental in mathematics, especially when demonstrating the relationship between two functions. For functions to be inverses of each other, each function should undo the other. This means if we apply one function and then its inverse, we should get back our original value. To show this for functions f(x) and g(x), we analyze their compositions: f(g(x)) and g(f(x)).

For instance, with f(x) = 3 - 4x and g(x) = (3 - x)/4, we first substitute g(x) into f(x) and then f(x) into g(x). The goal is to simplify the result to just x in both cases. Such simplifications lead to the conclusion that f(g(x)) = x and g(f(x)) = x, thus proving algebraically that f and g are indeed inverse functions. In essence, this verification confirms that each function reverses the effect of the other, satisfying the definition of inverse functions.

Graphical Illustration

A graphical illustration can often provide an intuitive understanding of concepts like inverse functions. When we plot the graphs of two functions that are purported inverses of each other, there is a distinctive pattern to look for. The defining characteristic is that their graphs should be symmetrical with respect to the line y = x.

If we consider our functions from the exercise, f(x) = 3 - 4x and g(x) = (3 - x)/4, plotting them on a Cartesian plane along with the line y = x should reveal this symmetry. The line y = x acts like a mirror, where each point on the graph of f coincides with a point on the graph of g, when reflected over this line. This visual representation complements the algebraic proof by giving a clear, visual confirmation that f and g are inverse functions.

Function Composition

The concept of function composition is like a mathematical choreography where two dances (functions) are combined to form a new sequence (composition). When dealing with inverse functions, we specifically look at how the composition of a function with its inverse brings us back to the starting point. This is analogous to performing a dance sequence and then reversing it exactly to return to where you began.

In symbolic terms, the composition f(g(x)) or g(f(x)) should equal x if f and g are inverses. Understanding this concept is crucial because it confirms the relationship between f and g beyond just an algebraic identity—it's evidence of their intrinsic undoing property in the framework of function composition. Knowing how to compose functions and interpret their compositions is an essential skill for unraveling the deeper relationships between mathematical operations.