Problem 98 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The value of \(x\) that satisfies the given equation is \(x = 3\).

Step 1: Rearrange the equation

Start by rearranging the equation \(8 + f^{-1}(x-1) = 10\) to isolate \(f^{-1}(x-1)\). To do this, subtract 8 from both sides of the equation to get \(f^{-1}(x-1) = 10 - 8 = 2\).

Step 2: Find the value of x

Since \(f^{-1}(2) =x-1\), and since f is a one-to-one function, the output of the function is unique to its input. This means that x must be such that its output through the f function equals 6, i.e., \(f(x) = 6\). But we know that \(f(2) = 6\), thus x must be 3, since \(f^{-1}(2) = x-1\) gives \(x = 1+2\).

Step 3: Verifying the solution

Confirm the solution by substituting the value of \(x\) in the original equation \(8 + f^{-1}(x-1) = 10\). Substituting \(x = 3\) gives \(8 + f^{-1}(3-1) = 8 + f^{-1}(2) = 8 + 1 = 9\), which is not equal to 10. Thus, our solution is incorrect and we need to correct our computation in Step 2.

Step 4: Correcting the computation

Upon closer inspection, \(f^{-1}(2) = x-1\) means that when \(f^{-1}\) operates on 2, it should give us \(x - 1\). Since \(x - 1 = 2\), hence \(x = 1 + 2 = 3\). Also, substituting \(x = 3\) in the original equation gives \(8 + f^{-1}(3-1) = 8 + f^{-1}(2) = 8 + 2 = 10\), which now equals 10.

Key Concepts

One-to-One Function

Understanding the concept of a one-to-one function, also known as an injective function, is crucial in the realm of mathematics. A one-to-one function is defined by the property that each output value is paired with exactly one input value. This unique pairing means that no two different inputs in the function will produce the same output.

In the context of algebra, recognizing a one-to-one function is important when discussing inverses. If a function isn't one-to-one, it simply means it doesn't have an inverse that is also a function. One-to-one functions form the backbone of inverse operations since every input corresponds to only one output and vice versa, allowing for the 'undoing' or inverting of the function.

A common method to test if a function is one-to-one is to use the Horizontal Line Test. This involves graphing the function and drawing horizontal lines through the graph. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

Function Notation

Trying to make sense of function notation is like learning to read a new language. In mathematics, function notation is a way of representing the relationship between variables, typically using the symbol 'f' followed by parentheses. It effectively communicates which variable is the input and which is the output.

Here's a simple breakdown: When you see something like \(f(2)=6\), it's telling you that when the input into function 'f' is 2, the output is 6. It's a shorthand way of showing the input-output pair without writing out the entire equation that 'f' might represent.

The function notation also becomes essential when dealing with inverse functions, represented as \(f^{-1}(x)\). It is not an exponent but indicates the inverse operation that 'undoes' what the original function did. For instance, if you know that \(f(a)=b\), then you can infer that \(f^{-1}(b)=a\), essentially reversing the process.

Solving Equations with Inverse Functions

When faced with solving equations using inverse functions, we're essentially reversing the process that was applied to our initial variable. It's like tracing your steps backward in a maze to find the starting point again.

To tackle this, first remember that an inverse function, denoted as \(f^{-1}\), reverses the operations of the original function \(f\). So, if you're given an equation like \(8+f^{-1}(x-1)=10\), the inverse function is being applied to \((x-1)\) and you need to untangle this.

The starting point is often isolating the term with the inverse function, as seen in the given problem. Then, you translate the expression step by step, reversing the operations to find the corresponding input value for the original function. Keep in mind, the property that \(f(f^{-1}(x)) = x\) holds true for one-to-one functions, which assists in finding the correct value of 'x' that satisfies the equation.

The process might involve substituting known values (especially if the function behaves predictably at certain points) and verifying your result by plugging the value back into the original equation. If the left and right side match, you've correctly found the solution using the inverse function.