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Chapter 1: Problem 18

Determine the domain of each of the following functions. Explain your answers. \(f(x)=4 \quad g(x)=3 x+5 \quad h(x)=\frac{x-1}{x-2}\) \(F(x)=x^{2}-4 \quad G(x)=\frac{x-1}{x^{2}-4} \quad H(x)=\sqrt{x-2}\)

Short Answer

Expert verified
Domains: \(f(x)\) and \(g(x)\): all real numbers; \(h(x)\): all real numbers except \(2\); \(F(x)\): all real numbers; \(G(x)\): all real numbers except \(2\) and \(-2\); \(H(x)\): \(x \geq 2\).

Step by step solution

01

- Determine the domain of f(x)=4

The function \(f(x) = 4\) is a constant function. Constant functions are defined for all real numbers. Thus, the domain of \(f(x)\) is all real numbers.
02

- Determine the domain of g(x)=3x+5

The function \(g(x) = 3x + 5\) is a linear function. Linear functions are also defined for all real numbers. Thus, the domain of \(g(x)\) is all real numbers.
03

- Determine the domain of h(x)=\frac{x-1}{x-2}

The function \(h(x) = \frac{x-1}{x-2}\) is a rational function. Rational functions are undefined where the denominator is zero. Solve for \(x\) when the denominator is zero: \(x - 2 = 0\Rightarrow x = 2\). Therefore, the domain of \(h(x)\) is all real numbers except \(x = 2\).
04

- Determine the domain of F(x)=x^{2}-4

The function \(F(x) = x^2 - 4\) is a polynomial function. Polynomial functions are defined for all real numbers. Thus, the domain of \(F(x)\) is all real numbers.
05

- Determine the domain of G(x)=\frac{x-1}{x^{2}-4}

The function \(G(x) = \frac{x-1}{x^2 - 4}\) is a rational function. The denominator \(x^{2} - 4\) can be factored as \((x - 2)(x + 2)\). Thus, the denominator is zero when \(x = 2\) or \(x = -2\). Therefore, the domain of \(G(x)\) is all real numbers except \(x = 2\) and \(x = -2\).
06

- Determine the domain of H(x)=\sqrt{x-2}

The function \(H(x) = \sqrt{x - 2}\) is a square root function. Square root functions are defined when the radicand is non-negative. Solve for \(x\) when the radicand is non-negative: \(x - 2 \geq 0 \ x \geq 2\). Thus, the domain of \(H(x)\) is \(x \geq 2\).

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Key Concepts

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

Constant Function
A constant function is one where the value of the function remains the same, regardless of the input value. For example, let's take the function
\(f(x) = 4\). No matter what value you plug in for \(x\), the output will always be 4.
This is because there are no variables affecting the output.
The domain of a constant function is all real numbers. There's no restriction on the input values, so you can use any real number as the input. Hence,
the function \(f(x) = 4\) is defined for all \(x \in (-\infty, \infty)\).
Linear Function
A linear function creates a straight line when plotted on a graph. It has the form \(g(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
For instance, consider the function \(g(x) = 3x + 5\). Here, the slope \(m\) is 3 and the y-intercept \(b\) is 5.

The domain of a linear function is all real numbers, just like constant functions. You can substitute any real number for \(x\),
and you will always get a valid output. Therefore,
the function \(g(x) = 3x + 5\) is defined for all \(x \in (-\infty, \infty)\).
Rational Function
Rational functions are ratios of two polynomials. They are generally written as \(h(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials.
The domain of a rational function is all real numbers except where the denominator \(Q(x)\) is zero since division by zero is undefined.
For example, consider \(h(x) = \frac{x-1}{x-2}\). Here \(Q(x) = x - 2\), which is zero when \(x = 2\). Thus, the domain of \(h(x)\) is all real numbers except \(x = 2\):
\(x \in (-\infty, 2) \cup (2, \infty)\).
Polynomial Function
Polynomial functions are expressions involving powers of \(x\). They have the form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\).
For example, \(F(x) = x^2 - 4\) is a polynomial function.
The domain of polynomial functions is all real numbers as polynomials are defined for every real number.
There are no restrictions or undefined points you need to worry about.
Thus, the function \(F(x) = x^2 - 4\) is defined for all \(x \in (-\infty, \infty)\).
Square Root Function
Square root functions are of the form \(H(x) = \sqrt{x - a}\) where the expression inside the root (called the radicand) must be non-negative.
For example, consider \(H(x) = \sqrt{x - 2}\). The radicand here is \(x - 2\) and it needs to be greater than or equal to zero.
Solving for \(x\) when the radicand is non-negative gives us \(x - 2 \geq 0\)
or \(x \geq 2\). Thus, the domain of \(H(x)\) is all real numbers \(x\) such that \(x \geq 2\): \(x \in [2, \infty)\)

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Most popular questions from this chapter

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