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Chapter 1: Problem 35

Finding Domains of Functions and Composite Functions find (a) \(f \circ g\) and (b) \(g\) of. Find the domain of each function and each composite function. $$f(x)=\sqrt{x+4}, \quad g(x)=x^{2}$$

Short Answer

Expert verified
The composite function \(f \circ g(x) = \sqrt{x^2+4}\) with a domain of all real numbers, and \(g \circ f(x) = (\sqrt{x+4})^2\) with a domain \(x \ge -4\). The domain of function \(f(x)\) is \(x \ge -4\) and for \(g(x)\), it's all real numbers.

Step by step solution

01

Find the Composite Functions

To form the composite functions, substitute the whole function \(g(x) = x^2\) into \(f(x)\) to obtain \(f \circ g(x)\) and vice versa for \(g \circ f(x)\). (a) The composition \(f \circ g(x) = f(g(x)) = f(x^2)\). To find the composite function, replace every \(x\) in \(f\) with \(g(x)\), thus the composite function \(f(g(x)) = \sqrt{x^2+4}\). (b) The composition \(g \circ f(x) = g(f(x)) = g(\sqrt{x+4})\). By replacing every \(x\) in \(g\) with \(f(x)\), the composite function is \(g(f(x)) = (\sqrt{x+4})^2\).
02

Find the Domains of Each Function.

The domain of a function is a set of all possible inputs (or \(x\) values) for which the function is defined. (a) For \(f(x) = \sqrt{x+4}\), the term inside the square root must be greater than or equal to 0 in order for \(f(x)\) to have real values. Hence, \(x+4 \ge 0\), the domain of \(f(x)\) is \(x \ge -4\). (b) For \(g(x) = x^2\), the function is defined for all values of \(x\). Hence its domain is all real numbers.
03

Find the Domains of Each Composite Function.

(a) For \(f \circ g(x) = \sqrt{x^2+4}\), the term inside the square root must be greater than or equal to 0 to produce real values. This holds for all \(x\), hence, the domain is all real numbers. (b) For \(g \circ f(x) = (\sqrt{x+4})^2\), the term under the square root in \(f(x)\) must be greater than or equal to 0. This gives \(x+4 \ge 0\) or \(x \ge -4\). Hence, the domain is \(x \ge -4\).

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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 concept that demarcates all possible input values or 'x' values that a function can accept. To determine the domain, we consider any restrictions due to mathematical operations within the function. For instance, with square root functions, the term within the root must be non-negative.

To find the domain:
  • Identify and handle any fractions to ensure the denominator does not equate to zero.
  • Check square root expressions to ensure terms under the root are zero or positive.
  • Consider logarithmic functions where the argument should be positive.
These steps ensure that we avoid undefined expressions, keeping the function behavior well-defined for all values within its domain.
Square Root Function
A square root function involves finding a number which, when multiplied by itself, yields the original number inside the square root. It's essential to note that square root functions typically produce non-negative numbers.

In mathematical terms, for a function like \(f(x) = \sqrt{x+4}\), the term inside the square root, \(x+4\), must be \(\geq 0\) to result in a real number. Thus, solving \(x+4 \geq 0\) gives the domain \(x \geq -4\).

When plotting, the graph of a square root function starts at the point \(x = -4\) and extends right, portraying an increasing curve. Recognizing these properties helps clarify why certain domains are chosen for functions involving square roots.
Function Composition
Function composition is a key method of combining functions where the output of one function becomes the input of another. This is denoted by \(f \circ g(x)\) or \(g \circ f(x)\). For instance, if \(f(x) = \sqrt{x+4}\) and \(g(x) = x^2\), we construct composites like \(f(g(x))\) (i.e., \(f(x^2)\)) which results in \(\sqrt{x^2 + 4}\).

Steps to form a composite function:
  • Substitute the function \(g(x)\) into every instance of \(x\) in \(f(x)\), or the reverse.
  • Simplify the resulting expressions to determine any necessary restrictions.
The domain of these composites depends on ensuring the inner function's domain aligns with making the outer function's expression valid as well.
Real Numbers
Real numbers are a collection of numbers encompassing rational and irrational numbers. This includes everything along a number line, from negative to positive infinity. In dealing with functions, particularly complex ones like compositions or roots, most domains are considered within the scope of real numbers as they provide a broad range of valid inputs.

Properties of real numbers:
  • Include integers, fractions, and irrationals like \(\pi\) and \(\sqrt{2}\).
  • Cover the entire continuous range with no gaps.
Functions like \(g(x) = x^2\) have domains that encompass all real numbers because squaring is valid for any real, providing a clear depiction of influence over outputs. Understanding them is crux to modeling actual-world phenomena or solving many algebraic functions involving domain restrictions.

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Most popular questions from this chapter

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