Problem 21 - Algebra Form and Function

Question

In Problems \(21-25,\) check that the functions are inverses. $$ f(x)=2 x-7 \text { and } g(t)=\frac{t}{2}+\frac{7}{2} $$

Step-by-Step Solution

Answer

Answer: Yes, these functions are inverses of each other. This is because the compositions of the functions, \(f(g(t)) = t\) and \(g(f(x)) = x\), are equal to the input variables.

Step 1: Composition of functions f(g(t))

First, we will find the composition of function f with function g: \(f(g(t))\). This means that we will plug the expression for g(t) into f(x). $$ f(g(t)) = f\left(\frac{t}{2} + \frac{7}{2}\right) $$

Step 2: Solve for f(g(t))

Now we substitute the expression for f(x) into our composition: $$ f(g(t)) = 2\left(\frac{t}{2} + \frac{7}{2}\right)-7 $$

Step 3: Simplify f(g(t))

Next, we'll simplify the expression we got in Step 2: $$ f(g(t)) = t + 7 - 7 $$ The result of the simplification is: $$ f(g(t)) = t $$

Step 4: Composition of functions g(f(x))

Now we'll find the composition of function g with function f: \(g(f(x))\). This means that we will plug the expression for f(x) into g(t). $$ g(f(x)) = g(2x - 7) $$

Step 5: Solve for g(f(x))

Now we substitute the expression for g(t) into our composition: $$ g(f(x)) = \frac{2x - 7}{2} + \frac{7}{2} $$

Step 6: Simplify g(f(x))

Next, we'll simplify the expression we got in Step 5: $$ g(f(x)) = x - \frac{7}{2} + \frac{7}{2} $$ The result of the simplification is: $$ g(f(x)) = x $$

Step 7: Check if both compositions are equal to the input variables

In the previous steps, we found that \(f(g(t)) = t\) and \(g(f(x)) = x\). Since both compositions equal to the input variables, we can conclude that functions f and g are inverses of each other.

Key Concepts

composition of functions

When dealing with inverse functions, the concept of composition is vital. Essentially, composing functions involves plugging one function into another. In the exercise, we have functions \(f(x)\) and \(g(t)\). To check if they are inverses, we first perform the composition \(f(g(t))\). This means substituting the expression of \(g(t)\) into \(f(x)\). Similarly, we compute \(g(f(x))\) by substituting \(f(x)\) into \(g(t)\). If both compositions return the input variable (\(f(g(t)) = t\) and \(g(f(x)) = x\)), the functions are inverse of each other.

• Step 1: Start with \(f(g(t)) = f\left(\frac{t}{2} + \frac{7}{2}\right)\).
• Step 4: Then move to \(g(f(x)) = g(2x - 7)\).

These steps confirm whether the given functions truly "undo" each other.

function simplification

Once you've substituted one function into the other, it's critical to simplify the resulting expression to see if it equals the input variable. Simplification helps reveal whether the composition resolves back to the original value, which is key in verifying inverse functions.For \(f(g(t))\):

• Insert \(g(t)\) into \(f(x)\), giving \(f(g(t)) = 2\left( \frac{t}{2} + \frac{7}{2} \right) - 7\).
• Simplify: Multiply and combine like terms to get \(f(g(t)) = t + 7 - 7\).
• Final result: \(f(g(t)) = t\).

For \(g(f(x))\):

• Insert \(f(x)\) into \(g(t)\), giving \(g(f(x)) = \frac{2x - 7}{2} + \frac{7}{2}\).
• Simplify: Combine fractions and like terms to reach \(g(f(x)) = x - \frac{7}{2} + \frac{7}{2}\).
• Final result: \(g(f(x)) = x\).

This simplification confirms that when composed, both functions reduce to their respective input variables.

algebraic expressions

Algebraic expressions are the foundation of composing and simplifying functions. Understanding how to handle expressions, such as fractions and variables, allows you to navigate through problems involving inverse functions smoothly.Here’s a quick reminder of some important algebraic rules:

• Distribute multiplication across terms in parentheses.
• Combine like terms to simplify expressions further.
• Cancel out terms (such as adding and subtracting the same value), which is crucial in simplifying compositions to see if \(f(g(t)) = t\) and \(g(f(x)) = x\).

The goal is always to manipulate the expressions so that they reveal the true relationship between the functions, demonstrating their inverse nature.