Problem 39 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=4 x+7 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x - 7}{4} \).

Step 1: Replace f(x) with y

Start by rewriting the function with y in place of f(x). This gives us the equation \( y = 4x + 7 \). We will solve for x to find the inverse.

Step 2: Solve for x

To find the inverse, we need to solve the equation \( y = 4x + 7 \) for x. Begin by subtracting 7 from both sides: \( y - 7 = 4x \).

Step 3: Isolate x

Divide both sides of the equation by 4 to solve for x. This yields \( x = \frac{y - 7}{4} \).

Step 4: Swap variables

To express the inverse function, swap x and y in the equation \( x = \frac{y - 7}{4} \). This gives us \( y = \frac{x - 7}{4} \), which is the inverse function.

Step 5: Write the inverse function

The inverse function \( f^{-1}(x) \) is \( \frac{x - 7}{4} \). We have successfully found the inverse of the given function.

Key Concepts

Function Notation

Function notation is a way of expressing the relationship between input and output variables in mathematical functions. It is a shorthand representation of a function's operation, allowing for concise communication of complex ideas. When using function notation, you generally see symbols like \(f(x)\) or \(g(x)\), where \(f\) indicates the function's name, and \(x\) is the input variable or independent variable.

• \(f(x) = 4x + 7\): This is the function notation for the given problem. Here, \((x)\) follows \(f\) indicating the input, and the expression \(4x + 7\) is what the function does to \(x\).
• Using function notation, finding an inverse involves switching the roles of \(x\) and \(y\), making the inputs and outputs swap places in the function.

Function notation helps simplify identifying functions and their operations within complex calculations, making it easier to manipulate and analyze the relationships between variables.

Linear Functions

Linear functions describe a straight-line relationship between two variables. They can be expressed in the form \(y = mx + c\), where \(m\) is the slope of the line, and \(c\) is the y-intercept. In the exercise example, the function \(f(x) = 4x + 7\) is linear, characterized by a slope of 4 and a y-intercept of 7.

• Slope (\(m\)): Determines the steepness of the line. In our case, for every one unit increase in \(x\), \(y\) increases by 4.
• Y-intercept (\(c\)): Indicates the point where the line crosses the y-axis. Here, when \(x = 0\), \(y\) is 7.

Linear functions are straightforward to work with because of their consistent rate of change. They provide foundational understanding for more complex functions and are essential in calculating inverse functions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and solving equations to isolate variables or find specific solutions. It is a core skill needed to solve for the inverse of a function, like in our example where we aim to find the inverse of \(f(x) = 4x + 7\).

To find an inverse:

• First, rewrite the original function by replacing \(f(x)\) with \(y\), giving \(y = 4x + 7\).
• Next, solve for \(x\) by performing algebraic operations: subtract 7 and then divide by 4 to isolate \(x\).
• Finally, swap \(x\) and \(y\) to express the inverse function as \(y = \frac{x - 7}{4}\).

Algebraic manipulation is crucial in solving equations and allows you to express relationships between variables in different forms, including moving from standard to inverse functions.