Chapter 1

If the composite functions \(f(g(x))\) and \(g(f(x))\) both equal \(x\), then the function \(g\) is the ________ function of \(f\).

Two functions \(f\) and \(g\) can be combined by the arithmetic operations of ________, ________, ________,and _________ to create new functions.

Horizontal shifts, vertical shifts, and reflections are called ________ transformations.

The graph of a function \(f\) is the collection of ________ ________ \((x, f(x))\) such that \(x\) is in the domain of \(f\).

A relation that assigns to each element \(x\) from a set of inputs, or ________, exactly one element \(y\) in a set of outputs, or ________, is called a ________.

In Exercises 1-7, fill in the blanks. The simplest mathematical model for relating two variables is the ________ equation in two variables \( y = mx + b \).

An ordered pair \( (a, b) \) is a __________ of an equation in \( x \) and \( y \) if the equation is true when \( a \) is substituted for \( x \), and \( b \) is substituted for \( y \).

Match each term with its definition. (a) \( x \)-axis (i) point of intersection of vertical axis and horizontal axis (b) \( y \)-axis (ii) directed distance from the \( x \)-axis (c) origin (iii) directed distance from the \( y \)-axis (d) quadrants (iv) four regions of the coordinate plane (e) \( x \)-coordinate (v) horizontal real number line (f) \( y \)-coordinate (vi) vertical real number line

Statisticians use a measure called ________ of________ ________ to find a model that approximates a set of data most accurately.

The inverse function of \(f\) is denoted by ________.

A reflection in the \(x\)-axis of \(y = f(x)\) is represented by \(h(x) = \) ________, while a reflection in the \(y\)-axis of \(y = f(x)\) is represented by \(h(x) = \) ________.

In Exercises 1-9, match each function with its name. \(f(x) = x\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

The ________ ________ ________ is used to determine whether the graph of an equation is a function of \(y\) in terms of \(x\).

Functions are commonly represented in four different ways, ________, ________, ________, and ________.

In Exercises 1-7, fill in the blanks. For a line, the ratio of the change in \( y \) to the change in \( x \) is called the ________ of the line.

The set of all solution points of an equation is the ________ of the equation.

In Exercises 2 - 4, fill in the blanks. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ______________ plane.

The linear model with the least sum of square differences is called the ________ ________ ________ line.

The domain of \(f\) is the ________ of \(f^{-1}\), and the ________ of \(f^{-1}\) is the range of \(f\).

The domain of \((f \circ g)\) is all \(x\) in the domain of \(g\) such that _______ is in the domain of \(f\).

Transformations that cause a distortion in the shape of the graph of \(y = f(x)\) are called ________ transformations.

In Exercises 1-9, match each function with its name. \(f(x) = 1/x\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

The ________ of a function \(f\) are the values of \(x\) for which \(f(x) = 0\).

For an equation that represents \(y\) as a function of \(x\), the set of all values taken on by the ________ variable \(x\) is the domain, and the set of all values taken on by the ________ variable is the range.

In Exercises 1-7, fill in the blanks. Two lines are ________ if and only if their slopes are equal.

The points at which a graph intersects or touches an axis are called the ________ of the graph.

An \(r\)-value of a set of data, also called a ________ ________, gives a measure of how well a model fits a set of data.

The graphs of \(f\) and \(f^{-1}\) are reflections of each other in the line ________.

A nonrigid transformation of \(y = f(x)\) represented by \(h(x) = f(cx)\) is a ________ ________ if \(c > 1\) and a ________ ________ if \(0 < c < 1\).

In Exercises 1-9, match each function with its name. \(f(x) = x^2\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

A function \(f\) is ________ on an interval if, for any \(x_1\) and \(x_2\) in the interval, \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\).

The function given by \( f(x) = \left\\{ \begin{array}{ll} 2x - 1, & \mbox{ \) x < 0 \(} \\ x^2 + 4, & \mbox{ \) x \ge 0 \(} \end{array} \right.\)

In Exercises 1-7, fill in the blanks. Two lines are ________ if and only if their slopes are negative reciprocals of each other.

A graph is symmetric with respect to the ________ if, whenever \( (x, y) \) is on the graph, \( (-x, y) \) is also on the graph.

Direct variation models can be described as "\(y\) varies directly as \(x\)," or "\(y\) is ________ ________ to \(x\)."

A function \(f\) is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable.

A nonrigid transformation of \(y = f(x)\) represented by \(g(x) = cf(x)\) is a ________ ________ if \(c > 1\) and a ________ ________ if \(0 < c < 1\).

In Exercises 1-9, match each function with its name. \(f(x) = \sqrt{x}\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

A function value \(f(a)\) is a relative ________ of \(f\) if there exists an interval \((x_1, x_2)\) containing \(a\) such that implies \(f(a) \geq f(x)\).

If the domain of the function \(f\) is not given, then the set of values of the independent variable for which the expression is defined is called the ________ ________.

The equation \( (x - h)^2 + (y - k)^2 = r^2 \) is the standard form of the equation of a ________ with center ________ and radius ________.

In direct variation models of the form \(y = kr\), \(k\) is called the ________ of ________.

A graphical test for the existence of an inverse function of \(f\) is called the _______ Line Test.

Match the rigid transformation of \(y=f(x)\) with the correct representation of the graph of \(h\), where \(c > 0\). (a) \(h(x) = f(x) + c\) (i) A horizontal shift of \(f\), \(c\) units to the right (b) \(h(x) = f(x) - c\) (ii) A vertical shift of \(f\), \(c\) units downward (c) \(h(x) = f(x + c)\) (iii) A horizontal shift of \(f\), \(c\) units to the left (d) \(h(x) = f(x - c)\) (iv) A vertical shift of \(f\), \(c\) units upward

In Exercises 1-9, match each function with its name. \(f(x) = c\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

In calculus, one of the basic definitions is that of a ________ ________, given by \(\frac{f(x+h)-f(x)}{h}\), \(h \neq 0\).

When you construct and use a table to solve a problem, you are using a ________ approach.

The direct variation model \(y = kr^n\) can be described as "\(y\) varies directly as the \(n\)th power of \(x\)," or "\(y\) is ________ ________ to the \(n\)th power of \(x\)."

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1} (x)) = x\) and \(f^{-1} (f (x)) = x\). \(f(x) = 6x\)

For each function, sketch (on the same set of coordinate axes) a graph of each function for \(c = -1\), \(1\), and \(3\). (a) \(f(x) = |x| + c\) (b) \(f(x) = |x - c|\) (c) \(f(x) = |x + 4| + c\)