Problem 62 - 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. $$g^{-1}(7)$$

Step-by-Step Solution

Answer

The output for the function \(g^{-1}(7)\) is 2

Step 1: Understand the Inverse Function

An inverse function is a function that undoes the work of the original function. In other words, if the function g takes a number 'x' and turns it into '7', the inverse function \(g^{-1}\) should take '7' and forms it back to 'x'. So, we can take our function g and set it equal to 7 since that is what \(g^{-1}(7)\) implies.

Step 2: Solve the Equation

We know the function g(x) = 4x - 1 and we need to solve g(x) = 7. Thus we get the equation 4x - 1 = 7. Solving this equation will provide us the value of 'x' which is the output of the inverse function.

Step 3: Find the Value of x

The equation 4x - 1 = 7 can be solved by first adding 1 to both sides, to give 4x = 7 + 1 which gives 4x = 8. Then divide each side by 4 to solve for x which gives \(x=g^{-1}(7)=2\)

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics where you take two functions and combine them to create a new function. Imagine each function as a machine that transforms an input into an output. When you compose functions, you feed the output of one machine directly into the input of another.

Function composition is usually represented as \( (f \circ g)(x) = f(g(x)) \). In this notation:

• First, you evaluate the inner function, which is \(g(x)\) in the example above.
• Next, you take the result obtained from \(g(x)\) and evaluate it using the outer function \(f(x)\).

This process allows for complex transformations of inputs by linking simpler transformations together.

Think of it as a way to "chain" functions together, creating a more elaborate process for transforming the initial input value. It's useful for managing complex operations, as it breaks them down into manageable parts.

Solving Linear Equations

Solving linear equations is an essential skill in algebra. Linear equations are equations of the first order, which means they involve only the first power of the variable. These equations can be written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

To solve a linear equation, you need to isolate the variable on one side of the equation. Here's a simple step-by-step process:

• First, look at the equation, such as \(4x - 1 = 7\).
• Next, eliminate any constants from the side of the equation with the variable. This is done by adding 1 to both sides, resulting in \(4x = 8\).
• Finally, divide both sides by the coefficient of the variable, which will give you \(x = 2\).

Solving linear equations is all about balancing and simplifying the equation until you successfully isolate the variable. This method not only applies to simple linear equations but can also be extended to solve more complex systems of equations.

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input value. Think of a function as a rule that assigns each input (commonly an x-value) to exactly one output (a y-value).

For example, if we have a function \(f(x) = 2x - 5\), and we want to evaluate this function at \(x = 3\), we substitute 3 into the function:

• Replace 'x' with 3 in the expression: \(f(3) = 2(3) - 5\).
• Calculate the result by performing the multiplication and subtraction: \(f(3) = 6 - 5 = 1\).

Function evaluation is a straightforward yet critical practice. It links abstract equations to concrete numbers and is essential for verifying solutions and understanding how functions behave with specific inputs.

This skill is vital when dealing with problems that require you to determine the inverse or when you need to validate the results.