Problem 68 - College Algebra

Question

If \(f(2)=6,\) find \(x\) satisfying \(8+f^{-1}(x-1)=10\)

Step-by-Step Solution

Answer

The solution for \(x\) is 7.

Step 1: Simplify the Equation

Firstly, simplify the equation \(8 + f^{-1}(x - 1) = 10\) by subtracting 8 from both sides. Thus, it gives \(f^{-1}(x - 1) = 2\).

Step 2: Identify the inverse function

Based on \(f(2) = 6\), and since the inverse function undoes the action of the function, it implied that \(f^{-1}(6) = 2\). Considering this and the equation from step 1, we can replace \(f^{-1}(x - 1) = 2\) with \(x - 1 = 6\).

Step 3: Calculate 'x'

Finally, rearrange \(x - 1 = 6\) by adding 1 to both sides. This gives \(x = 6 + 1 = 7\).

Key Concepts

Function Notation

Function notation is a way to represent the relationship between input and output values in functions clearly. It is a fundamental concept in algebra and is expressed generally as \(f(x)\), where \(f\) is the function and \(x\) is the input value.
In this form, \(f(x)\) indicates the output derived when \(x\) is the input.

• For example, if \(f(2) = 6\), it means when \(x\) is 2, the output, or \(f(x)\), is 6.
• Similarly, inverse function notation, \(f^{-1}(x)\), represents the reverse operation. If \(f(x)\) transforms \(x\) to a certain \(y\), \(f^{-1}(y)\) converts \(y\) back to \(x\).

Understanding function notation is crucial for the setup and solving of equations where input and output transformations occur, allowing for clear tracking of these changes.

Solving Equations

Solving equations involves finding values for unknown variables that make the equation true. Understanding how to simplify expressions and transform equations is key.
In the provided exercise, start by simplifying the expression \(8 + f^{-1}(x - 1) = 10\).

• Subtract 8 from both sides to isolate the inverse function: \(f^{-1}(x - 1) = 2\).
• This operation ensures the equation is in a simpler form, making it easier to analyze and solve.
• Be precise in each step to avoid errors, and always verify your solution by substituting back into the original equation.

Solving equations often requires patience. But with practice and attention to detail, it becomes a manageable and rewarding process.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to isolate variables. It is essential in finding solutions to equations.
In our problem, once we determined \(f^{-1}(x - 1) = 2\), we use algebraic manipulation to find \(x\).

• Replace the inverse function by its known value: \(x - 1 = 6\).
• To isolate \(x\), add 1 to both sides of the equation: \(x - 1 + 1 = 6 + 1\), resulting in \(x = 7\).

This systematic approach to dealing with expressions ensures clarity and accuracy. By consistently applying algebraic rules, you build a strong foundation for tackling more complex mathematical problems.