Problem 9 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. $$ \begin{array}{l}{g(x)=3 x-2} \\ {f(x)=\frac{x-2}{3}}\end{array} $$

Step-by-Step Solution

Answer

The functions are not inverse functions.

Step 1: Finding the Composition of Functions

To determine if two functions \(f(x)\) and \(g(x)\) are inverses, you need to verify whether \(f(g(x)) = x\) and \(g(f(x)) = x.\) Start with evaluating \(f(g(x))\). Substitute \(g(x) = 3x - 2\) into \(f(x):\)\[f(g(x)) = f(3x-2) = \frac{3x-2-2}{3} = \frac{3x-4}{3} = x - \frac{4}{3}.\]

Step 2: Verify the First Composition Equals x

After calculating \(f(g(x))\), we get \(x - \frac{4}{3}\), which is not equal to \(x\). Therefore, \(f(g(x)) eq x.\)

Step 3: Finding the Second Composition

Now, evaluate \(g(f(x))\). Substitute \(f(x) = \frac{x-2}{3}\) into \(g(x):\)\[g(f(x)) = g\left(\frac{x-2}{3}\right) = 3\left(\frac{x-2}{3}\right) - 2 = (x-2) - 2 = x - 4.\]

Step 4: Verify the Second Composition Equals x

After calculating \(g(f(x))\), we get \(x - 4\), which is not equal to \(x\). Therefore, \(g(f(x)) eq x.\) Both compositions \(f(g(x))\) and \(g(f(x))\) do not yield \(x\), confirming the functions are not inverses.

Key Concepts

Composition of Functions

Understanding the composition of functions is key when determining if two functions are inverse functions. It involves substituting one function into another. This creates a new function that combines the effects of both functions. For example:

• Start with two functions, \(f(x)\) and \(g(x)\).
• Compose them by replacing the variable in the first function with the second function. This is expressed as \(f(g(x))\).

To check if two functions are inverses, compose them in both possible orders:

• Calculate \(f(g(x))\) and see if it simplifies to \(x\).
• Then calculate \(g(f(x))\) and verify if it also simplifies to \(x\).

If both compositions equal \(x\), then the functions are indeed inverses of each other.

Function Verification

Function verification is essential to solidify that two functions are true inverses. After computing the compositions \(f(g(x))\) and \(g(f(x))\), each result must equal \(x\). If either result deviates from \(x\), then the functions are not inverses. Let's review the steps:

• Calculate \(f(g(x))\). In this case, it led to \(x - \frac{4}{3}\), a result not equal to \(x\).
• Next, verify \(g(f(x))\), which resulted in \(x - 4\), also not equal to \(x\).

Both results must match exactly with \(x\). If not, these results confirm that the original functions do not act as inverses. Verify by checking your calculations carefully, as algebraic mistakes can lead to incorrect conclusions.

Algebraic Manipulation

Algebraic manipulation plays a crucial role in simplifying expressions when working with functions. It requires rearranging equations and simplifying them to examine relationships between variables.

• Take the expression \(f(g(x)) = \frac{3x-4}{3}\). Simplify it carefully, watching out for algebraic errors.
• Then, review \(g(f(x)) = x - 4\). Such manipulations can either confirm or refute if the composed functions yield \(x\).

Algebraic manipulation can be improved by:

• Double-checking calculations and simplifying step by step.
• Understanding operations like distributing and combining like terms.
• Practicing with different equations to become familiar with common patterns.

A small mistake in this phase can drastically change the results, so it’s vital to proceed methodically.