Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 83
Use the two given functions to write \(y\) as a function of \(x\). $$y=w^{2}-2, w=x+3$$
Short Answer
Expert verified
y = x^2 + 6x + 7
Step by step solution
01
- Identify the given functions
We have two given functions: 1. The first function is 饾惒 as a function of 饾惏, which is given by 饾惒 = 饾惏^2 - 2.2. The second function is 饾惏 as a function of 饾惐, which is given by 饾惏 = 饾惐 + 3.
02
- Substitute the second function into the first
Substitute 饾惏 = 饾惐 + 3 into the 饾惒 function. This means we replace 饾惏 in 饾惒 = 饾惏^2 - 2 with 饾惐 + 3.
03
- Perform the substitution
After substitution, the function becomes: 饾惒 = (饾惐 + 3)^2 - 2.
04
- Simplify the expression
Expand (饾惐 + 3)^2 to get 饾惐^2 + 6饾惐 + 9, then subtract 2. The simplified expression is: 饾惒 = 饾惐^2 + 6饾惐 + 9 - 2.
05
- Finalize the function
Combine like terms to arrive at the final function: 饾惒 = 饾惐^2 + 6饾惐 + 7.
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.
Function Composition
Function composition is a fundamental concept in mathematics where you apply one function to the results of another function. It's like putting one machine inside another and watching what comes out.
In the exercise, you start with two functions:
In the exercise, you start with two functions:
- y as a function of w, given by
y = w^2 - 2. - w as a function of x, given by
w = x + 3.
To find y as a function of x, you substitute the second function into the first one. This process is called 'composing' the functions. So, you replace w in y = w^2 - 2 with x + 3.
This produces: y = (x + 3)^2 - 2.
< Combining functions in this way lets you connect different processes, making it easier to work with complex expressions.
Combining functions in this way lets you connect different processes, making it easier to work with complex expressions.
Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
For example, y = x^2 + 6x + 7 is a polynomial function.
Polynomial functions are used widely across different fields because they can model a wide range of real-world scenarios.
For example, y = x^2 + 6x + 7 is a polynomial function.
Polynomial functions are used widely across different fields because they can model a wide range of real-world scenarios.
- They can describe the path of a projectile.
- Model economic growth.
- Even represent population changes over time.
Solving polynomial functions often requires algebraic manipulation - simplifying and combining like terms, factoring, or applying the quadratic formula. By understanding polynomial functions, you gain the ability to work with many types of equations you'll encounter in math and science.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with. It鈥檚 like untangling a knot to make things clear.
In our problem, after substituting w with (x + 3), we get: (x + 3)^2 - 2.
We need to expand this expression: (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9.
Then, we subtract 2 to get:
y = x^2 + 6x + 9 - 2.
Simplifying this gives:
y = x^2 + 6x + 7.
Algebraic manipulation helps in simplifying expressions, solving equations, and arriving at a more understandable form of the function.
In our problem, after substituting w with (x + 3), we get: (x + 3)^2 - 2.
We need to expand this expression: (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9.
Then, we subtract 2 to get:
y = x^2 + 6x + 9 - 2.
Simplifying this gives:
y = x^2 + 6x + 7.
Algebraic manipulation helps in simplifying expressions, solving equations, and arriving at a more understandable form of the function.
Simplifying Expressions
Simplifying expressions refers to reducing them to their simplest form. It makes equations easier to work with and understand.
In the given exercise, after expanding and combining like terms, we went from y = (x + 3)^2 - 2 to y = x^2 + 6x + 7.
Key steps include:
In the given exercise, after expanding and combining like terms, we went from y = (x + 3)^2 - 2 to y = x^2 + 6x + 7.
Key steps include:
-
Expanding expressions (x + 3)(x + 3) to x^2 + 6x + 9. - Combining like terms (adding up coefficients of x).
- Simplifying constants (9 - 2).
Mastering simplification involves practice and understanding of algebraic rules. It helps to spot patterns and essential elements within complex expressions and guides towards the solution in a systematic way.
Simplifying is critical in algebra as it frequently used in other areas of mathematics to make equation solving more manageable.
