Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer.The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Step-by-Step Solution

Answer

Yes, the statement is true. The graphs of the functions \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical as \(|-x|\) and \(|x|\) represent the same value, denoting the same distance from zero, regardless of the sign of the number within the absolute value.

Step 1: Understand the functions

The given functions are \(f(x)=|x|+6\) and \(f(x)=|-x|+6\). The first function could be broken down as \(f(x) = |x| + 6\), meaning it represents the distance of \(x\) from 0 on the number line, shifted 6 units upwards. The second function \(f(x)=|-x|+6\) also represents the distance of \(-x\) from 0 on the number line, shifted 6 units upwards.

Step 2: Understand the absolute value

Let's understand the behavior of the absolute value function. The absolute value of a number is its distance from 0 on the number line, regardless of the direction. Hence, \(|-x|\) is equivalent to \(|x|\), because either way we are measuring the distance of the number from 0.

Step 3: Comparison of the functions

Comparing \(f(x)=|x|+6\) and \(f(x)=|-x|+6\), we can see, since \(|-x|\) and \(|x|\) has the same value, the two functions are indeed identical.

Key Concepts

Graphing Functions

A function's graph is a visual representation of the relationship between input values (often represented as "x") and output values ("y"). For absolute value functions, the graph typically forms a "V" shape. Let's explore the graph of an absolute value function using the example function, \(f(x)=|x|+6\). - The \(|x|\) part of the function creates a "V" with its point at the origin (0,0), extending outwards symmetrically.- The "+6" shifts the entire graph upward by 6 units, so the vertex of the "V" is now at (0, 6).
By graphing \(f(x)=|x|+6\), we see two rays: one rising from (0,6) towards the right, and the other extending from (0,6) to the left, creating the characteristic "V" shape.
The graph of \(f(x)=|-x|+6\) will look identical because, as we will see, \(|-x|\) is equal to \(|x|\). Thus, even when the function is written as \(f(x)=|-x|+6\), the shape and position of the graph remain unchanged.

Function Transformation

Transforming functions involves shifting, stretching, or reflecting graphs. Absolute value functions can be uniquely transformed by modifying their expressions. In our example, consider the absolute value functions \(f(x)=|x|+6\) and \(f(x)=|-x|+6\).- **Vertical Shifts**: Adding a number, like "+6" in both functions, shifts the graph vertically. Positive numbers shift it up, and negative numbers shift it down.- **Reflections**: Normally, changing \(x\) to \/(-x\/ doesn't affect absolute value functions because \(|-x|\) equals \(|x|\). Thus, there is no horizontal reflection.
Regardless of whether \(x\) is replaced with \(-x\), the function is unchanged due to the nature of absolute value. Only vertical shifts change the graph visually. This simplifies understanding transformations for absolute value functions.

Equivalent Functions

Equivalent functions produce the same output for every possible input. When comparing \(f(x)=|x|+6\) and \(f(x)=|-x|+6\), equivalency comes from the properties of absolute values.- Since \( |x| = |-x| \), replacing \( x \) with \( -x \) does not change the function's value.- Both functions have an identical expression other than the input sign, plus the identical constant of "+6" added.
No matter what x-value you substitute, both functions yield identical results. Therefore, they are equivalent. Their graphs are identical, verifying that transforming \(x\) to \(-x\) in an absolute value function doesn't alter its fundamental nature. Understanding this key property can simplify analyzing and graphing many absolute value functions.