Functions

Function Notation

Intermediate Algebra for College Students

determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I must have made a mistake in finding the composite functions $f \circ g$ and $g \circ f,$ because I notice that $f \circ g$ is the same function as $g \circ f$

Exponential and Logarithmic Functions

Composite and Inverse Functions

00:28

Intermediate Algebra for College Students

use a graphing utility to graph each
function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one).
$f(x)=\frac{x^{3}}{2}$

Exponential and Logarithmic Functions

Composite and Inverse Functions

01:15

Intermediate Algebra for College Students

The graph represents the probability that two people in the same room share a birthday as a function of the number of people in the room. Call the function $f$
a. Explain why $f$ has an inverse that is a function.
b. Describe in practical terms the meanings of $f^{-1}(0.25)$ $f^{-1}(0.5),$ and $f^{-1}(0.7)$

Exponential and Logarithmic Functions

Composite and Inverse Functions

Domain and Range

946 Practice Problems

00:52

A Graphical Approach to Precalculus with Limits

State the open intervals over which each function is (a) increasing.
(b) decreasing, and (c) constant.
$y=-\frac{2}{3}|x+3|-4$ (See FIGURE 41.)

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

00:44

A Graphical Approach to Precalculus with Limits

Let the domain of $f(x)$ be [-1,2] and the range be $[0,3] .$ Find the domain and range of the following.
$$-f(x)$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

01:00

A Graphical Approach to Precalculus with Limits

Let the domain of $f(x)$ be [-1,2] and the range be $[0,3] .$ Find the domain and range of the following.
$$f(x-2)$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

One-to-One vs Subjective Function

135 Practice Problems

01:53

Precalculus

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form y = ƒ -11x2, (b) graph ƒ and ƒ -1 on the same axes, and (c) give the domain and the range of ƒ and ƒ -1. If the function is not one-to-one, say so.
$$f(x)=\frac{-3 x+12}{x-6}, \quad x \neq 6$$

Equations and Inequalities

Linear Equations

02:05

Precalculus

Equations and Inequalities

Linear Equations

01:16

Precalculus : Building Concepts and Connections

State whether each function is one-to-one.
$$f(x)=-2 x^{3}+4$$

Exponential and Logarithmic Functions

Inverse Functions

Rates of Change

115 Practice Problems

04:24

Calculus: Early Transcendental Functions

To start skating, you must angle your foot and push off the ice. Alain Haché's The Physics of Hockey derives the relationship between the skate angle $\theta,$ the sideways stride distance $s,$ the stroke period $T$ and the forward specd $v$ of the skater, with $\theta=\tan ^{-1}\left(\frac{2 y}{r^{2}}\right) .$ For $T=1$ second, $s=60 \mathrm{cm}$ and an acceleration of $1 \mathrm{m} / \mathrm{s}^{2}$, find the rate of change of the angle $\theta$ when the skater reaches (a) $1 \mathrm{m} / \mathrm{s}$ and (b) $2 \mathrm{m} / \mathrm{s}$. Interpret the sign and size of $\theta^{\prime}$ in terms of skating technique.(FIGURE CANNOT COPY)

Applications of Differentiation

Related rates

02:13

Calculus: Early Transcendental Functions

Suppose that the average yearly cost per item for producing $x$ items of a business product is $\bar{C}(x)=10+\frac{100}{x} .$ If the current production is $x=10$ and production is increasing at a rate of 2 items per year, find the rate of change of the average cost.

Applications of Differentiation

Related rates

03:56

Calculus: Early Transcendental Functions

It can be shown that solutions of the logistic equation have the form $p(t)=\frac{B}{1+A e^{-k t}},$ for constants $B, A$ and $k .$ Find the rate of change of the population and find the limiting population, that is, $\lim _{t \rightarrow \infty} p(t)$

Applications of Differentiation

Rates of Change in Economics and The Sciences

Function Composition

463 Practice Problems

04:45

Intermediate Algebra for College Students

The regular price of a computer is $x$ dollars. Let $f(x)=x-400$ and $g(x)=0.75 x$
a. Describe what the functions $f$ and $g$ model in terms of the price of the computer.
b. Find $(f \circ g)(x)$ and describe what this models in terms of the price of the computer.
c. Repeat part (b) for $(g \circ f)(x)$.
Which composite function models the greater discount on the computer, $f \circ g$ or $g \circ f ?$ Explain.
e. Find $f^{-1}$ and describe what this models in terms of the price of the computer.

Exponential and Logarithmic Functions

Composite and Inverse Functions

00:31

Intermediate Algebra for College Students

let
\[
\begin{array}{l}
f(x)=2 x-5 \\
g(x)=4 x-1 \\
h(x)=x^{2}+x+2
\end{array}
\]
Evaluate the indicated function without finding an equation for the function.
$(f \circ g)(0)$

Exponential and Logarithmic Functions

Composite and Inverse Functions

00:27

Intermediate Algebra for College Students

use the graphs of $f$ and $g$ to evaluate each
composite function.
$(f \circ g)(-1)$

Exponential and Logarithmic Functions

Composite and Inverse Functions

Transformation of Functions and Their Graphs

401 Practice Problems

00:46

A Graphical Approach to Precalculus with Limits

the figure shows a transformation of the graph of $y=|x| .$ Write the equation for the graph. Refer to Example 7 as needed.

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

02:23

A Graphical Approach to Precalculus with Limits

Use transformations of graphs to sketch a graph of $y=f(x)$ by hand.
$$f(x)=|-2 x+1|$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

01:46

A Graphical Approach to Precalculus with Limits

Use transformations of graphs to sketch a graph of $y=f(x)$ by hand.
$$f(x)=|-x+3|$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

Absolute Value Functions and Their Graphs

198 Practice Problems

02:03

A Graphical Approach to Precalculus with Limits

Use transformations of graphs to sketch a graph of $y=f(x)$ by hand.
$$f(x)=|-x-2|$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

00:55

A Graphical Approach to Precalculus with Limits

Use transformations of graphs to sketch a graph of $y=f(x)$ by hand.
$$f(x)=|x+2|-3$$

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

01:03

A Graphical Approach to Precalculus with Limits

the graph of $y=f(x)$ has been transformed to the graph of $y=g(x)$ No shrinking or stretching is involved. Give the equation of $y=g(x)$.

Analysis of Graphs of Functions

Stretching, Shrinking, and Reflecting Graphs

Radical Functions and Their Graphs

358 Practice Problems

02:53

Intermediate Algebra for College Students

Graph $f(x)=\sqrt{(x-1)^{2}}$ by hand.

Radicals, Radical Functions, and Rational Exponents

Multiplying and Simplifying Radical Expressions

02:08

Intermediate Algebra for College Students

Your cardiac index is your heart's output, in liters of blood per minute, divided by your body's surface area, in square meters.
The cardiac index, $C(x),$ can be modeled by
\[C(x)=\frac{7.644}{\sqrt[4]{x}}, \quad 10 \leq x \leq 80\]
where $x$ is an individual's age, in years. The graph of the function is shown. Use the function to solve.
a. Find the cardiac index of a 32 -year-old. Express the denominator in simplified radical form and reduce the fraction.
b. Use the form of the answer in part (a) and a calculator to express the cardiac index to the nearest hundredth. Identify your solution as a point on the graph.

Radicals, Radical Functions, and Rational Exponents

Multiplying and Simplifying Radical Expressions

02:41

Intermediate Algebra for College Students

Paleontologists use the function
\[W(x)=4 \sqrt{2 x}\]
to estimate the walking speed of a dinosaur, $W(x),$ in feet per second, where $x$ is the length, in feet, of the dinosaur's leg. The graph of $W$ is shown in the figure. Use this information to solve.
What is the walking speed of a dinosaur whose leg length is 6 feet? Use the function's equation and express the answer in simplified radical form. Then use the function's graph to estimate the answer to the nearest foot per second.

Radicals, Radical Functions, and Rational Exponents

Multiplying and Simplifying Radical Expressions

Inverse Functions and Their Graphs

418 Practice Problems

00:25

Algebra and Trigonometry

Use the values listed in Table 6 to evaluate or solve.
$$\text { Solve } f^{-1}(x)=7$$

Functions

Inverse Functions

00:24

Algebra and Trigonometry

Evaluate or solve, assuming that the function $f$ is one-to-one.
$$\text { If } f(6)=7, \text { find } f^{-1}(7)$$

Functions

Inverse Functions

01:07

Algebra and Trigonometry

Use the graph of $f$ shown in Figure 11.
$$\text { Find } f^{-1}(0)$$

Functions

Inverse Functions

Direct Variation Problems

157 Practice Problems

01:27

Precalculus

Find the constant of variation " $k$ " and write the variation equation, then use the equation to solve.
The weight of an object on the moon varies directly with the weight of the object on Earth. A $96-$ kg object on Earth would weigh only $16 \mathrm{kg}$ on the moon. How much would a fully suited $250-\mathrm{kg}$ astronaut weigh on the moon?

Polynomial and Rational Functions

Variation: Function Models in Action

04:41

Precalculus

Find the constant of variation and write the related variation equation. Then use the equation to complete the table or solve the application.
$C$ varies directly with $R$ and inversely with $S$ squared, and $C=21$ when $R=7$ and $S=1.5$.
$$\begin{array}{|c|c|c|}\hline R & S & C \\\hline 120 & & 22.5 \\\hline 200 & 12.5 & \\\hline & 15 & 10.5 \\
\hline\end{array}$$

Polynomial and Rational Functions

Variation: Function Models in Action

02:34

Precalculus

Find the constant of variation and write the variation equation. Then use the equation to complete the table or solve the application.
$p$ varies directly with the square of $q ; p=280$ when $q=50$
$$\begin{array}{|c|c|}\hline q & p \\\hline 45 & \\\hline & 338.8 \\\hline 70 & \\\hline\end{array}$$

Polynomial and Rational Functions

Variation: Function Models in Action

Indirect Variation Problems

108 Practice Problems

00:56

Precalculus

Write the variation equation for each statement.
Horsepower varies jointly as the number of cylinders in the engine and the square of the cylinder's diameter.

Polynomial and Rational Functions

Variation: Function Models in Action

02:07

Precalculus: Graphs and Models

The surface area $S$ of a sphere varies directly as the square of its radius $r .$ What happens to the area if the radius is cut in half?

Polynomial and Rational Functions

Variation and Modeling

01:51

Precalculus: Graphs and Models

The frequency of vibration fof a musical string is directly proportional to the square root of the tension $T$ and inversely proportional to the length $L$ of the string. If the tension of the string is increased by a factor of 4 and the length of the string is doubled, what is the cffcct on the frequency?

Polynomial and Rational Functions

Variation and Modeling

Even-odd functions

25 Practice Problems

01:43

Algebra and Trigonometry

Show that if $f$ is any function, then the function $O$ defined by
$$O(x)=\frac{f(x)-f(-x)}{2}$$
is odd.

Even-odd functions

0 Practice Problems

Even-odd functions

0 Practice Problems