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Chapter 2: Problem 53

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x^{2}, g(x)=\sqrt{1-x}$$

Short Answer

Expert verified
(a) (f 鈭� g)(x) = 1 - x, domain: (-鈭�, 1]; (b) (g 鈭� f)(x) = 鈭�(1 - x虏), domain: [-1, 1]; (c) (f 鈭� f)(x) = x鈦�, domain: (-鈭�, 鈭�).

Step by step solution

01

Find the Composite Function (f 鈭� g)(x)

The composition \((f \circ g)(x)\) means \(f(g(x))\). First, find \(g(x)\): \(g(x) = \sqrt{1-x}\). Now substitute \(g(x)\) into \(f(x)\):\[f(g(x)) = f(\sqrt{1-x}) = (\sqrt{1-x})^2 = 1-x\].
02

Determine the Domain of (f 鈭� g)(x)

The function \((f \circ g)(x) = 1-x\) must respect the domain of \(g(x) = \sqrt{1-x}\). The expression under the square root must be non-negative: \[1-x \geq 0 \Rightarrow x \leq 1\].Thus, the domain of \((f \circ g)(x)\) is \( (-\infty, 1] \).
03

Find the Composite Function (g 鈭� f)(x)

The composition \((g \circ f)(x)\) means \(g(f(x))\). First, find \(f(x)\): \(f(x) = x^2\). Now substitute \(f(x)\) into \(g(x)\):\[g(f(x)) = g(x^2) = \sqrt{1-x^2}\].
04

Determine the Domain of (g 鈭� f)(x)

The function \((g \circ f)(x) = \sqrt{1-x^2}\) must have the expression under the square root non-negative:\[1-x^2 \geq 0 \Rightarrow x^2 \leq 1\]. Solving gives \(-1 \leq x \leq 1\).Thus, the domain of \((g \circ f)(x)\) is \([-1, 1]\).
05

Find the Composite Function (f 鈭� f)(x)

The composition \((f \circ f)(x)\) means \(f(f(x))\). First, find \(f(x)\): \(f(x) = x^2\). Now substitute \(f(x)\) into itself:\[f(f(x)) = f(x^2) = (x^2)^2 = x^4\].
06

Determine the Domain of (f 鈭� f)(x)

The function \((f \circ f)(x) = x^4\) has no restrictions on \(x\) since any real number can be squared or raised to a higher even power.Thus, the domain of \((f \circ f)(x)\) is \( (-\infty, \infty) \).

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Key Concepts

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

Domain of a Function
The domain of a function is a super important concept in understanding how functions work. It refers to all the possible input values (usually represented as \(x\)) that a function can accept without causing any issues. This ensures the function will calculate a valid output. Imagine the domain as a filter for inputs that are allowed to flow into the function.

Here鈥檚 what you need to consider when determining the domain:
  • For square root functions like \(g(x) = \sqrt{1-x}\), the expression inside the square root must be non-negative because the square root of a negative number is not real.
  • For polynomial functions like \(f(x) = x^2\), the domain is generally unrestricted, which means \((-\infty, \infty)\) because any real number can be squared.
  • When composing functions (like finding \((f \circ g)(x)\)), it鈥檚 crucial to consider the domains of each individual function to determine the overall domain of the composed function.
Being able to identify the domain helps in avoiding undefined operations, like division by zero or taking the square root of a negative number, ensuring that you can work with the function smoothly.
Square Root Function
A square root function is a special type of function that involves the square root of an expression. It鈥檚 often represented as \(g(x) = \sqrt{1-x}\) in the original exercise.

Key points about square root functions:
  • The expression under the square root sign must be non-negative. This is because square roots of negative numbers are not defined in the real number system.
  • To find the domain of a square root function like \(\sqrt{1-x}\), you must solve the inequality \(1-x \geq 0\). This ensures that the expression inside the square root is zero or positive.
  • Applying this specific rule, we can calculate the domain of \(\sqrt{1-x}\) as \((-\infty, 1]\). This set of \(x\) values keeps the square root value real and valid.
By understanding these aspects, evaluating where the square root function is defined becomes straightforward.
Polynomial Function
Polynomial functions are among the most common and versatile functions you'll encounter in mathematics. They are expressed as sums of terms consisting of a variable raised to a non-negative integer power, along with coefficients. In the given exercise, \(f(x) = x^2\) is a polynomial function.

Here's what makes polynomial functions special:
  • They have a domain of all real numbers: \( (-\infty, \infty) \). This means you can input any real number into the function and get a valid output.
  • The smooth and continuous nature of polynomial functions makes them easy to work with, as there are no abrupt changes or undefined points within their scope of definition.
  • When composing polynomial functions with other types (like the square root function in the example), it鈥檚 crucial to focus on the domains and ensure each step remains valid within its constraints.
Polynomial functions provide a solid foundation in math due to their simplicity and broad applicability.

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