Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 67
Graph. $$ y=\left|2^{x^{7}}-1\right| $$
Short Answer
Expert verified
The graph of y = |2^{x^7} - 1| involves two parts: y = 2^{x^7} - 1 for x > 0, and y = 1 - 2^{x^7} for x < 0, meeting at x = 0.
Step by step solution
01
- Identify the function
The given function is y = |2^{x^7} - 1|. This is an absolute value function involving an exponential term.
02
- Consider the inner function
Focus on the expression inside the absolute value: 2^{x^7} - 1. Determine where this expression is positive, negative, or zero.
03
- Determine when the inner function equals zero
Solve for when 2^{x^7} - 1 = 0. This occurs when 2^{x^7} = 1. Since 2 to any power is 1 only when that power is 0, we have x^7 = 0. Thus, x = 0.
04
- Evaluate behavior of the inner function
For x > 0, 2^{x^7} grows exponentially, making 2^{x^7} - 1 positive. Conversely, for x < 0, 2^{x^7} still grows, but x^7 allows it to approach zero, making 2^{x^7} - 1 negative.
05
- Apply absolute value
Use the absolute value property to handle positive and negative parts. For x > 0, y = 2^{x^7} - 1. For x < 0, y = -(2^{x^7} - 1) = 1 - 2^{x^7}. At x = 0, y = 0.
06
- Graph the function
Plot y = 2^{x^7} - 1 for x > 0, and y = 1 - 2^{x^7} for x < 0. These two parts come together smoothly at x = 0. The plot for y should have a sharp V-shape at x=0 and be flat relative to large positive/negative x .
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function
An exponential function involves a constant base raised to a variable exponent, commonly written as \( a^x \), where \( a \) is a positive constant. For example, in the function \( 2^{x^7} \), the base is 2 and the exponent is \( x^7 \). Exponential functions are crucial because they grow (increase or decrease) extremely fast.
Key properties include:
Key properties include:
- When the exponent is positive, the function's value increases very rapidly as \( x \) increases.
- When the exponent is negative, the function's value approaches zero as \( x \) decreases.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For instance, \( |3| = 3 \) and \( |-3| = 3 \).
The absolute value function, \( |f(x)| \), changes how functions behave by ensuring the output is non-negative.
Consider our exercise function \( y = |2^{x^7} - 1| \):
The absolute value function, \( |f(x)| \), changes how functions behave by ensuring the output is non-negative.
Consider our exercise function \( y = |2^{x^7} - 1| \):
- If \( 2^{x^7} - 1 \geq 0 \): \( y = 2^{x^7} - 1 \)
- If \( 2^{x^7} - 1 < 0 \): \( y = -(2^{x^7} - 1) = 1 - 2^{x^7} \)
Function Graphing
Graphing functions involves plotting points on the coordinate plane to represent the function鈥檚 output for different inputs. For complex functions like \( y = |2^{x^7} - 1| \), follow these steps:
For this function:
- Identify key points where the behavior changes, such as when the inner function = 0.
- Analyze the behavior of the function in different intervals, one for \( x > 0 \) and one for \( x < 0 \).
For this function:
- For \( x > 0 \), plot \( y = 2^{x^7} - 1 \).
- For \( x < 0 \), plot \( y = 1 - 2^{x^7} \).
- The function meets smoothly at \( x = 0 \) with \( y = 0 \).
Intermediate Algebra
Intermediate algebra forms the bridge between basic algebra and more advanced concepts like calculus. It introduces more complex functions, including exponential and absolute value functions.
Important topics include:
Important topics include:
- Manipulating algebraic expressions.
- Understanding function properties and transformations.
- Graphing different types of functions, like polynomial, exponential, and piecewise functions.
