Problem 30 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. $$ \begin{array}{l}{f(x)=x-5} \\ {g(x)=x+5}\end{array} $$

Step-by-Step Solution

Answer

Yes, the functions are inverse functions.

Step 1: Understanding the Concept

Two functions, \(f(x)\) and \(g(x)\), are inverses if \(f(g(x)) = x\) and \(g(f(x)) = x\). This means applying one function and then the other should return the original input.

Step 2: Find \(f(g(x))\)

Substitute \(g(x) = x+5\) into \(f(x) = x-5\):\[ f(g(x)) = f(x+5) = (x+5) - 5 = x \]This simplifies to \(x\), confirming the condition \(f(g(x)) = x\).

Step 3: Find \(g(f(x))\)

Substitute \(f(x) = x-5\) into \(g(x) = x+5\):\[ g(f(x)) = g(x-5) = (x-5) + 5 = x \]This also simplifies to \(x\), confirming the condition \(g(f(x)) = x\).

Step 4: Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), the functions \(f(x) = x-5\) and \(g(x) = x+5\) are inverse functions.

Key Concepts

Functions

Functions are an essential part of mathematics, particularly in algebra and calculus. They represent a relationship between sets of input and output values, usually expressed as \(y = f(x)\). Here, \(x\) is the input value, and \(y\) is the output value. You can think of a function as a machine that takes an input, performs a specific operation, and then provides an output based on that operation.

Each function has a unique output associated with every input. This means that if a function is given the same input value twice, it will always produce the same output. This predictability is what makes functions such a powerful mathematical concept. For functions to be useful, they must also pass the vertical line test. This test states that if a vertical line crosses the graph of the function more than once, then the relation is not a function.

In our exercise, we have two functions: \(f(x) = x - 5\) and \(g(x) = x + 5\). Understanding the mechanism of how these function works is foundational to examining if they are inverses of each other.

Composition of Functions

Composition of functions involves applying one function to the results of another function. If you have two functions, say \(f(x)\) and \(g(x)\), composing them means creating a new function by substituting one function into the other. This can be mathematically represented as \(f(g(x))\) or \(g(f(x))\).

The compositions \(f(g(x))\) and \(g(f(x))\) give us powerful insight into the relationship between the two functions. If performing a composition results in the original input, this indicates a special kind of relationship - namely, that the functions are inverses of each other.

• When \(f(g(x)) = x\), it means starting with an input \(x\), applying \(g\), and then \(f\) takes us back to \(x\).
• Conversely, if \(g(f(x)) = x\), beginning with \(x\), using \(f\), followed by \(g\), also returns the initial value \(x\).

Understanding these compositions allows us to definitively determine if two functions are inverse functions as demonstrated in the exercise where both compositions resulted in \(x\).

Algebra

Algebra helps us understand how functions work and the relationships between them. It allows us to manipulate equations and simplify expressions, which is crucial in solving problems involving inverse functions.

In the context of inverse functions, using algebra helps in writing and simplifying expressions for compositions such as \(f(g(x))\) and \(g(f(x))\). For example, substituting \(g(x) = x + 5\) into \(f(x) = x - 5\), we perform algebraic operations to simplify:

• Start with \(f(g(x)) = f(x+5)\)
• Simplify \(f(x+5) = (x+5)-5\)
• This leads to \(x\), showing that \(f(g(x)) = x\)

A similar process confirms \(g(f(x)) = x\).

By mastering these algebraic techniques, you gain the ability to explore various properties of functions and investigate their inverses, improving your overall problem-solving skills in mathematics. This forms a solid foundation for tackling more advanced topics in algebra and calculus.