Problem 61 - College Algebra Essentials

Question

Let $$\begin{aligned}&f(x)=2 x-5\\\&g(x)=4 x-1\\\&h(x)=x^{2}+x+2\end{aligned}$$ Evaluate the indicated function without finding an equation for the function. $$f^{-1}(1)$$

Step-by-Step Solution

Answer

Thus, \( f^{-1}(1) = 3 \).

Step 1: Identify the target value

First, recognize that we need to find a value \( a \) such that \( f(a) = 1 \).

Step 2: Set up an equation

Next, set up the equation \( 2a - 5 = 1 \) based on the function definition of \( f(x) \). This equation represents all values \( x = a \) for which \( f(x) = 1 \).

Step 3: Solve for a

Add 5 to both sides of the equation to get \( 2a = 6 \). Then, divide both sides by 2 to solve for \( a \), yielding \( a = 3 \).

Key Concepts

Algebraic Functions

In mathematics, algebraic functions play a fundamental role in various calculations and problem-solving scenarios. An algebraic function is made up of a combination of constants, variables, and arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation with integers as exponents. These functions are expressed using algebraic expressions, which can be simple linear relationships like f(x) = 2x - 5 or more complex ones including polynomials, like h(x) = x^2 + x + 2.

A key aspect of algebraic functions is their representation in the form of equations that describe how one variable (usually y or f(x)) depends on another variable (x). Through algebraic manipulation, we can transform and analyze these functions to better understand their behavior and the relationships they depict.

For the inverse function, denoted as f^{-1}(x), it is essentially the 'undoing' operation. For the function f(x) = 2x - 5, its inverse would answer the question: What value of x do we input into f to get a specific output value? In other words, it reverses the roles of the input and the output.

Function Evaluation

Evaluation of a function, such as f(x), involves substituting the variable x with a specific value to calculate the corresponding output of the function. This concept is tightly linked with algebraic functions, where calculations often require us to plug in numbers or expressions to find out what the function 'returns' for those specific inputs.

When we evaluate the inverse function, f^{-1}(1) for example, we're looking for the original input X that would give us the output of 1. This is an essential skill, as it helps us to not only verify solutions but also understand the practical implications of mathematical models in real-world situations. When given f(x)=2x-5, evaluating the function is straightforward, but working backward to find the inverse requires us to set this expression equal to a given output and solve for the input that would lead to this result.

Equations Solving

Solving equations is at the heart of mathematics, especially when dealing with functions. The process typically involves finding the value(s) of the variable(s) that make the equation true. When we work with algebraic functions, solving these equations can range from simple arithmetic to more advanced algebraic techniques.

In our example, we aim to find the value that makes f(x) = 1. We do this by rearranging the equation 2a - 5 = 1 to solve for 'a'. This involves a series of steps such as adding 5 to both sides and then dividing by 2, which gives us the solution a = 3. This value of 'a' is crucial as it represents the input we seek for the inverse function. It's also worth mentioning that not all algebraic functions will have an inverse that is itself a function, especially if the original function isn't one-to-one (each x value maps to exactly one y value). Occasionally, additional steps such as restricting the domain of the original function may be necessary to ensure the inverse also functions properly.