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Chapter 1: Problem 3

What is the domain of a rational function?

Short Answer

Expert verified
Question: Determine the domain of the following rational function: R(x) = \frac{4x-5}{x^2 - 16}. Answer: To find the domain of the rational function, we follow the steps outlined in the solution. 1. Identify the denominator Q(x) of the rational function: Q(x) = x^2 - 16 2. Set the denominator equal to zero (Q(x) = 0) and solve for x: x^2 - 16 = 0 (x-4)(x+4) = 0 The solutions are x = 4 and x = -4. 3. The domain of the rational function is all real numbers except the x-values that make Q(x) = 0. Therefore, the domain of the rational function R(x) = \frac{4x-5}{x^2 - 16} is all real numbers except x = 4 and x = -4.

Step by step solution

01

Defining a Rational Function

A rational function is a function that can be expressed as the quotient or fraction of two polynomials (P(x) and Q(x)) with Q(x) not equal to zero. It can be written as: R(x) = \frac{P(x)}{Q(x)}
02

Understanding Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of x-values for which the function produces a valid output or value.
03

Determining the Domain of a Rational Function

To determine the domain of a rational function, we need to find all x-values that make the denominator (Q(x)) equal to zero, since division by zero is undefined in mathematics. 1. Identify the denominator Q(x) of the rational function. 2. Set the denominator equal to zero (Q(x) = 0) and solve for x. 3. The domain of the rational function is all real numbers except the x-values that make Q(x) = 0.
04

Example

Let's find the domain of a sample rational function: R(x) = \frac{2x + 3}{x^2 -9} The denominator of the rational function is Q(x) = x^2 - 9. Now we set Q(x) equal to zero and solve for x: x^2 - 9 = 0 (x-3)(x+3) = 0 The solutions are x = 3 and x = -3. Therefore, the domain of the rational function R(x) is all real numbers except x = 3 and x = -3.

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Most popular questions from this chapter

a. If \(f(0)\) is defined and \(f\) is an even function, is it necessarily true that \(f(0)=0 ?\) Explain. b. If \(f(0)\) is defined and \(f\) is an odd function, is it necessarily true that \(f(0)=0 ?\) Explain.

The height in feet of a baseball hit straight up from the ground with an initial velocity of \(64 \mathrm{ft} / \mathrm{s}\) is given by \(h=f(t)=64 t-16 t^{2},\) where \(t\) is measured in seconds after the hit. a. Is this function one-to-one on the interval \(0 \leq t \leq 4 ?\) b. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels upward. Express your answer in the form \(t=f^{-1}(h)\) c. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels downward. Express your answer in the form \(t=f^{-1}(h)\) d. At what time is the ball at a height of \(30 \mathrm{ft}\) on the way up? e. At what time is the ball at a height of \(10 \mathrm{ft}\) on the way down?

Find a polynomial \(f\) that satisfies the following properties. (Hint: Determine the degree of \(f ;\) then substitute a polynomial of that degree and solve for its coefficients.) $$f(f(x))=9 x-8$$

Relative acuity of the human eye The fovea centralis (or fovea) is responsible for the sharp central vision that humans use for reading and other detail- oriented eyesight. The relative acuity of a human eye, which measures the sharpness of vision, is modeled by the function $$R(\theta)=\frac{0.568}{0.331|\theta|+0.568}$$ ,where \(\theta\) (in degrees) is the angular deviation of the line of sight from the center of the fovea (see figure). a. Graph \(R,\) for \(-15 \leq \theta \leq 15\) b. For what value of \(\theta\) is \(R\) maximized? What does this fact indicate about our eyesight? c. For what values of \(\theta\) do we maintain at least \(90 \%\) of our maximum relative acuity? (Source: The Journal of Experimental Biology, 203, Dec 2000)

Find all the inverses associated with the following functions and state their domains. $$f(x)=2 /\left(x^{2}+2\right)$$
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