Problem 21 Find $(f \circ g)(x)$ and \((g... [FREE SOLUTION]

Chapter 12: Problem 21

Find $(f \circ g)(x)$ and $(g \circ f)(x) .$ $$ f(x)=|x| ; g(x)=10 x-3 $$

Short Answer

Expert verified

$(f \circ g)(x) = |10x - 3|$; $(g \circ f)(x) = 10|x| - 3$.

Step by step solution

Understanding the Problem

We are given two functions: $f(x)=|x|$ and $g(x)=10x-3$. We need to find the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$. This means we will substitute one function into the other.

Finding $(f \circ g)(x)$

The expression $(f \circ g)(x)$ means we apply $g(x)$ first and then $f$ to the result. Substitute $g(x)$ into $f(x)$: \[ (f \circ g)(x) = f(g(x)) = f(10x - 3) = |10x - 3|. \]This results in the absolute value of $10x - 3$.

Finding $(g \circ f)(x)$

The expression $(g \circ f)(x)$ means we apply $f(x)$ first and then $g$ to the result. Substitute $f(x)$ into $g(x)$: \[ (g \circ f)(x) = g(f(x)) = g(|x|) = 10|x| - 3. \]Here we replace $x$ with $|x|$ in $g(x)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function

The absolute value function is one of the fundamental concepts in algebra.
It is represented as $ f(x) = |x| $. This notation signifies the non-negative value of $x$. The absolute value essentially 'strips away' any negativity by transforming negative inputs into positive outputs, while leaving positive inputs unchanged.

For example, $|3| = 3$ and $|-3| = 3$.
This function graphically looks like a 'V' shape with its point at the origin $(0,0)$.

The absolute value is important because it allows us to measure distances accurately, without regard to direction on the number line.
In function composition, such as $ (f \circ g)(x) = |10x - 3| $, the role of the absolute value is to ensure the output reflects the magnitude of the expression $10x - 3$ regardless of its sign.

Intermediate Algebra

Intermediate algebra acts as a bridge between basic algebra concepts like solving linear equations and more complex concepts like polynomials.
It covers a wide array of topics that are foundational for advanced studies in mathematics.

Topics include working with rational expressions, radicals, and quadratic equations.
This level of algebra also introduces the concept of functions and their compositions.

Understanding function composition is crucial in intermediate algebra, as it helps in grasping how different functions interact with each other.
For example, in the original problem, we use composition to combine two functions, $ f(x) = |x| $ and $ g(x) = 10x - 3 $.
This illustrates how intermediate algebra provides tools to see the bigger picture in mathematical relationships.

Algebraic Functions

Algebraic functions are a class of functions that include operations of addition, subtraction, multiplication, division, and the extraction of roots from polynomials.
These functions often form the 'building blocks' in mathematics and can be combined in various ways to model real-world situations.

They can be expressed in the form $ f(x) = ax^n + bx^{n-1} + \ldots + c $, where $ a, b, c $ are constants and $ n $ is a positive integer.
A simple algebraic function like $ g(x) = 10x - 3 $ is linear, as it consists of a polynomial of degree 1.

When dealing with function compositions such as $(g \circ f)(x) = 10|x| - 3$, you are essentially nesting one algebraic function within another.
This creates more complex relationships and showcases the power and elegance of algebraic functions in mathematical modeling.

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91影视

Find \((f \circ g)(x)\) and \((g \circ f)(x) .\) $$ f(x)=|x| ; g(x)=10 x-3 $$

Short Answer

Step by step solution

Understanding the Problem

Finding \((f \circ g)(x)\)

Finding \((g \circ f)(x)\)

Key Concepts

Absolute Value Function

Intermediate Algebra

Algebraic Functions

One App. One Place for Learning.

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