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Chapter 1: Problem 138

Find \(f_{o} g\), where \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}-1\).

Short Answer

Expert verified
The composition of the functions \(f\) and \(g\), represented as \(f_{o} g\), is \(\sqrt{x^{2}-1}\).

Step by step solution

01

Identify the Functions

First, identify the functions involved in the composition. Here, we have functions \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}-1\).
02

Understand the Composition of Functions

The composition of functions \(f_{o} g\) is given by \(f(g(x))\). This implies that \(g(x)\) is substituted into \(f\).
03

Substitute g(x) into f(x)

Now, substitute \(g(x)=x^{2}-1\) into the function \(f(x)\). This gives \(f(g(x))=\sqrt{g(x)}=\sqrt{x^{2}-1}\). Hence, the composition \(f_{o} g\) is \(\sqrt{x^{2}-1}\).

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Key Concepts

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

Square Root Function
The square root function, represented by \( f(x) = \sqrt{x} \), is a fundamental mathematical concept. This function takes the square root of a given input value, \( x \). The result is a number which, when multiplied by itself, gives the original input value. For example, if \( x = 4 \), then \( f(x) = \sqrt{4} = 2 \), because \( 2^2 = 4 \).
Key properties of the square root function include:
  • Domain: The set of all non-negative numbers \( x \geq 0 \). This is because the square root of a negative number is not a real number.
  • Range: Non-negative numbers as well, \( f(x) \geq 0 \). This is because any square root is non-negative in the realm of real numbers.
  • Graph: The graph of \( f(x) = \sqrt{x} \) is a half-parabola opening to the right.
Understanding how to deal with the square root function is crucial for solving problems involving composition of functions, as it demands attention to the domain of input values.
Polynomial Functions
Polynomial functions are expressions involving terms in the form of \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer. The function \( g(x) = x^2 - 1 \) is a quadratic polynomial, which is a specific type of polynomial function.
Here's why polynomial functions like this are important:
  • Simplicity: Polynomials are straight-forward to analyze and compute, making them great building blocks for more complicated expressions.
  • Versatility: They can model a wide variety of real-world scenarios, from motion to growth rates.
  • Behavior: At the roots of the polynomial, the function value is zero. \( x^2 - 1 \) has roots at \( x = 1 \) and \( x = -1 \).
When considering polynomial functions like \( x^2 - 1 \), recognize how each term contributes to the behavior and shape of the function's graph. This understanding assists in applications like function composition, where polynomial outputs become inputs to other functions, such as the square root function in our exercise.
Mathematical Problem Solving
Mathematical problem solving, especially with function composition, requires a structured approach. In the problem, the task was to find the composition of two functions: \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \). This approach breaks down into several steps:
  • Identify and understand the functions: Before composing, recognize the functions involved and their distinct domains and ranges.
  • Understand the concept of function composition: Composition \( (f \circ g)(x) \) means substituting \( g(x) \) into \( f(x) \). This involves replacing \( x \) in \( f(x) \) with the expression for \( g(x) \).
  • Substitute and simplify: Here, substitute \( g(x) = x^2 - 1 \) into \( \sqrt{x} \), resulting in \( \sqrt{x^2 - 1} \).
It's essential to comprehend how each function and their characteristics affect the overall composition. Attention to mathematical details ensures correct solutions and strengthens problem-solving skills.

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Most popular questions from this chapter

If \(f(2+x)=f(2-x)\) and \(f(7-x)=f(7+x)\) and \(f(0)=0\), find the minimum number of roots of \(f(x)=0\), where \(20 \leq x \leq 20 .\)

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