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Module 1: Functions and Graphs

More function types, learning outcomes.

  • Identify a rational function
  • Describe the graphs of power and root functions
  • Explain the difference between algebraic and transcendental functions
  • Graph a piecewise-defined function

Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials. For example,

are rational functions. A root function is a power function of the form [latex]f(x)=x^{1/n}[/latex], where [latex]n[/latex] is a positive integer greater than one. For example, [latex]f(x)=x^{1/2}=\sqrt{x}[/latex] is the square-root function and [latex]g(x)=x^{1/3}=\sqrt[3]{x}[/latex] is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, [latex]f(x)=\sqrt{4-x^2}[/latex] is an algebraic function.

Example: Finding Domain and Range for Algebraic Functions

For each of the following functions, find the domain and range.

  • [latex]f(x)=\dfrac{3x-1}{5x+2}[/latex]
  • [latex]f(x)=\sqrt{4-x^2}[/latex]

When we multiply both sides of this equation by [latex]5x+2[/latex], we see that [latex]x[/latex] must satisfy the equation

From this equation, we can see that [latex]x[/latex] must satisfy

If [latex]y=\frac{3}{5}[/latex], this equation has no solution. On the other hand, as long as [latex]y\ne \frac{3}{5}[/latex],

These two inequalities reduce to [latex]2 \ge x[/latex] and [latex]x \ge -2[/latex]. Therefore, the set [latex]\{x|-2\le x\le 2\}[/latex] must be part of the domain. For both terms to be negative, we need

If [latex]-2 \le x \le 2[/latex], then [latex]0 \le 4-x^2 \le 4[/latex]. Therefore, [latex]0 \le \sqrt{4-x^2} \le 2[/latex], and the range of [latex]f[/latex] is [latex]\{y|0 \le y \le 2\}[/latex].

Find the domain and range for the function [latex]f(x)=\dfrac{(5x+2)}{(2x-1)}[/latex].

The denominator cannot be zero. Solve the equation [latex]y=\frac{(5x+2)}{(2x-1)}[/latex] for [latex]x[/latex] to find the range.

The domain is the set of real numbers [latex]x[/latex] such that [latex]x \ne \frac{1}{2}[/latex]. The range is the set [latex]\{y|y \ne \frac{5}{2}\}[/latex].

The root functions [latex]f(x)=x^{1/n}[/latex] have defining characteristics depending on whether [latex]n[/latex] is odd or even. For all even integers [latex]n \ge 2[/latex], the domain of [latex]f(x)=x^{1/n}[/latex] is the interval [latex][0,\infty)[/latex]. For all odd integers [latex]n \ge 1[/latex], the domain of [latex]f(x)=x^{1/n}[/latex] is the set of all real numbers. Since [latex]x^{1/n}=(−x)^{1/n}[/latex] for odd integers [latex]n, \, f(x)=x^{1/n}[/latex] is an odd function if [latex]n[/latex] is odd. See the graphs of root functions for different values of [latex]n[/latex] in Figure 11.

An image of two graphs. The first graph is labeled “a” and has an x axis that runs from -2 to 9 and a y axis that runs from -4 to 4. The first graph is of two functions. The first function is “f(x) = square root of x”, which is a curved function that begins at the origin and increases. The second function is “f(x) = x to the 4th root”, which is a curved function that begins at the origin and increases, but increases at a slower rate than the first function. The second graph is labeled “b” and has an x axis that runs from -8 to 8 and a y axis that runs from -4 to 4. The second graph is of two functions. The first function is “f(x) = cube root of x”, which is a curved function that increases until the origin, becomes vertical at the origin, and then increases again after the origin. The second function is “f(x) = x to the 5th root”, which is a curved function that increases until the origin, becomes vertical at the origin, and then increases again after the origin, but increases at a slower rate than the first function.

Figure 11. (a) If [latex]n[/latex] is even, the domain of [latex]f(x)=\sqrt[n]{x}[/latex] is [latex][0,\infty)[/latex]. (b) If [latex]n[/latex] is odd, the domain of [latex]f(x)=\sqrt[n]{x}[/latex] is [latex](-\infty,\infty )[/latex] and the function [latex]f(x)=\sqrt[n]{x}[/latex] is an odd function.

Example: Finding Domains for Algebraic Functions

For each of the following functions, determine the domain of the function.

  • [latex]f(x)=\dfrac{3}{x^2-1}[/latex]
  • [latex]f(x)=\dfrac{2x+5}{3x^2+4}[/latex]
  • [latex]f(x)=\sqrt{4-3x}[/latex]
  • [latex]f(x)=\sqrt[3]{2x-1}[/latex]
  • You cannot divide by zero, so the domain is the set of values [latex]x[/latex] such that [latex]x^2-1 \ne 0[/latex]. Therefore, the domain is [latex]\{x|x \ne \pm 1\}[/latex].
  • You need to determine the values of [latex]x[/latex] for which the denominator is zero. Since [latex]3x^2+4 \ge 4[/latex] for all real numbers [latex]x[/latex], the denominator is never zero. Therefore, the domain is [latex](-\infty,\infty )[/latex].
  • Since the square root of a negative number is not a real number, the domain is the set of values [latex]x[/latex] for which [latex]4-3x \ge 0[/latex]. Therefore, the domain is [latex]\{x|x \le \frac{4}{3}\}[/latex].
  • The cube root is defined for all real numbers, so the domain is the interval [latex](-\infty, \infty)[/latex].

Find the domain for each of the following functions: [latex]f(x)=\dfrac{(5-2x)}{(x^2+2)}[/latex] and [latex]g(x)=\sqrt{5x-1}[/latex].

Determine the values of [latex]x[/latex] when the expression in the denominator of [latex]f[/latex] is nonzero, and find the values of [latex]x[/latex] when the expression inside the radical of [latex]g[/latex] is nonnegative.

The domain of [latex]f[/latex] is [latex](-\infty, \infty)[/latex] The domain of [latex]g[/latex] is [latex]\{x|x \ge \frac{1}{5}\}[/latex].

Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are [latex]\sin x,\, \cos x, \, \tan x, \, \cot x,\, \sec x[/latex], and [latex]\csc x[/latex]. (We discuss trigonometric functions later in the module.) An exponential function is a function of the form [latex]f(x)=b^x[/latex], where the base [latex]b>0, \, b \ne 1[/latex]. A logarithmic function is a function of the form [latex]f(x)=\log_b(x)[/latex] for some constant [latex]b>0, \, b \ne 1[/latex], where [latex]\log_b(x)=y[/latex] if and only if [latex]b^y=x[/latex]. (We also discuss exponential and logarithmic functions later in the module.)

Example: Classifying Algebraic and Transcendental Functions

Classify each of the following functions, a. through c., as algebraic or transcendental.

  • [latex]f(x)= \dfrac{\sqrt{x^3+1}}{4x+2}[/latex]
  • [latex]f(x)=2^{x^2}[/latex]
  • [latex]f(x)=\sin (2x)[/latex]
  • Since this function involves basic algebraic operations only, it is an algebraic function.
  • This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
  • As in part (b), this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.

Watch the following video to see the worked solution to Example: Classifying Algebraic and Transcendental Functions

Is [latex]f(x)=\dfrac{x}{2}[/latex] an algebraic or a transcendental function?

Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function . The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of [latex]x[/latex]:

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for [latex]x<a[/latex] and [latex]x>a[/latex], we need to pay special attention to what happens at [latex]x=a[/latex] when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at [latex]x=a[/latex]. We examine this in the next example. If you need a refresher, check out the Recall box first.

Recall: Given a piecewise function, sketch a graph.

  • Indicate on the [latex]x[/latex]-axis the boundaries defined by the intervals on each piece of the domain.
  • For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

Example: Graphing a Piecewise-Defined Function

Sketch a graph of the following piecewise-defined function:

[latex]f(x)=\begin{cases} x+3, & x < 1 \\ (x-2)^2 & x \ge 1 \end{cases}[/latex]

Graph the linear function [latex]y=x+3[/latex] on the interval [latex](-\infty,1)[/latex] and graph the quadratic function [latex]y=(x-2)^2[/latex] on the interval [latex][1,\infty )[/latex]. Since the value of the function at [latex]x=1[/latex] is given by the formula [latex]f(x)=(x-2)^2[/latex], we see that [latex]f(1)=1[/latex]. To indicate this on the graph, we draw a closed circle at the point [latex](1,1)[/latex]. The value of the function is given by [latex]f(x)=x+2[/latex] for all [latex]x<1[/latex], but not at [latex]x=1[/latex]. To indicate this on the graph, we draw an open circle at [latex](1,4)[/latex].

An image of a graph. The x axis runs from -7 to 5 and the y axis runs from -4 to 6. The graph is of a function that has two pieces. The first piece is an increasing line that ends at the open circle point (1, 4) and has the label “f(x) = x + 3, for x < 1”. The second piece is parabolic and begins at the closed circle point (1, 1). After the point (1, 1), the piece begins to decrease until the point (2, 0) then begins to increase. This piece has the label “f(x) = (x - 2) squared, for x >= 1”.The function has x intercepts at (-3, 0) and (2, 0) and a y intercept at (0, 3).

Figure 12. This piecewise-defined function is linear for [latex]x<1[/latex] and quadratic for [latex]x \ge 1[/latex].

Sketch a graph of the function

[latex]f(x)=\begin{cases} 2-x, & x \le 2 \\ x+2, & x>2 \end{cases}[/latex]

Graph one linear function for [latex]x \le 2[/latex] and then graph a different linear function for [latex]x>2[/latex].

An image of a graph. The x axis runs from -6 to 5 and the y axis runs from -2 to 7. The graph is of a function that has two pieces. The first piece is a decreasing line that ends at the closed circle point (2, 0) and has the label “f(x) = 2 - x, for x <= 2. The second piece is an increasing line and begins at the open circle point (2, 4) and has the label “f(x) = x + 2, for x > 2.The function has an x intercept at (2, 0) and a y intercept at (0, 2).

Figure 13. Graph of piecewise function.

Example: Parking Fees Described by a Piecewise-Defined Function

In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional $2 for each hour or part thereof up to a maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight.

  • Write a piecewise-defined function that describes the cost [latex]C[/latex] to park in the parking garage as a function of hours parked [latex]x[/latex].
  • Sketch a graph of this function [latex]C(x)[/latex].
  • Since the parking garage is open 18 hours each day, the domain for this function is [latex]\{x|0 < x \le 18\}[/latex]. The cost to park a car at this parking garage can be described piecewise by the function [latex]C(x)=\begin{cases} \\ 10, & 0 < x \le 1 \\ 12, & 1 < x \le 2 \\ 14, & 2 < x \le 3 \\ 16, & 3 < x \le 4 \\ & \vdots \\ 30, & 10 < x \le 18 \end{cases}[/latex]

An image of a graph. The x axis runs from 0 to 18 and is labeled “x, hours”. The y axis runs from 0 to 32 and is labeled “y, cost in dollars”. The function consists 11 pieces, all horizontal line segments that begin with an open circle and end with a closed circle. The first piece starts at x = 0 and ends at x = 1 and is at y = 10. The second piece starts at x = 1 and ends at x = 2 and is at y = 12. The third piece starts at x = 2 and ends at x = 3 and is at y = 14. The fourth piece starts at x = 3 and ends at x = 4 and is at y = 16. The fifth piece starts at x = 4 and ends at x = 5 and is at y = 18. The sixth piece starts at x = 5 and ends at x = 6 and is at y = 20. The seventh piece starts at x = 6 and ends at x = 7 and is at y = 22. The eighth piece starts at x = 7 and ends at x = 8 and is at y = 24. The ninth piece starts at x = 8 and ends at x = 9 and is at y = 26. The tenth piece starts at x = 9 and ends at x = 10 and is at y = 28. The eleventh piece starts at x = 10 and ends at x = 18 and is at y = 30.

Figure 14. Graph of parking fees vs. hours spent parked in garage.

Watch the following video to see the worked solution to Example: Parking Fees Described by a Piecewise-Defined Function

The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is [latex]49\text{¢}[/latex] for the first ounce and [latex]21\text{¢}[/latex] for each additional ounce. Write a piecewise-defined function describing the cost [latex]C[/latex] as a function of the weight [latex]x[/latex] for [latex]0 < x \le 3[/latex], where [latex]C[/latex] is measured in cents and [latex]x[/latex] is measured in ounces.

The piecewise-defined function is constant on the intervals [latex](0,1], \, (1,2], \, \cdots[/latex]

[latex]C(x)=\begin{cases} 49, & 0 < x \le 1 \\ 70, & 1 < x \le 2 \\ 91, & 2 < x \le 3 \end{cases}[/latex]

  • 1.2 Basic Classes of Functions. Authored by : Ryan Melton. License : CC BY: Attribution
  • Calculus Volume 1. Authored by : Gilbert Strang, Edwin (Jed) Herman. Provided by : OpenStax. Located at : https://openstax.org/details/books/calculus-volume-1 . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike . License Terms : Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction

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Integrated math 1

Course: integrated math 1   >   unit 8.

  • Unit test Functions

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  1. Unit 2: More Function Types Flashcards

    a second-degree polynomial whose graph is a parabola. parabola. the U-shaped graph of a quadratic function. x-intercept. a point where a function touches or crosses the x-axis. axis of symmetry. the vertical line that separates the graph of a quadratic function into two halves that are reflections of each other.

  2. Solved Graded AssignmentUnit Test, Part 2: More Function

    Algebra questions and answers. Graded AssignmentUnit Test, Part 2: More Function TypesAnswer the questions below. When you are finished, submit this test to your teacher for full credit.Total score: , of 15 points (Score for Question 1: q, of 5 points)Graph the function f (x)=x-23.Answer:f (x) (C) 2016 K12 Inc.

  3. Unit 2: More Function Types Flashcards

    a family of functions of the form f (x) = C / 1 + Ae^-Bx, where A, B, and C are positive numbers. parent function. the basic function in a family of functions. piecewise function. a function that is defined using different rules for different intervals of the domain. Unit 2: More Function Types.

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  9. Functions (part 2): Unit test

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  18. Unit Test: Using Function Models

    The function f (x) is the total amount spent at a store, when purchasing x items that are $5 each and the items are not taxable. What is the practical domain for the function f (x)? all whole numbers. Consider two functions: g (x)=20 (1.5)x and the function f (x) shown in the table. x f (x) −5 −45. −4 −48. −3 −49.

  19. PDF Unit Test, Part 2: Quadratic Functions

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