Chapter 1

Fill in the blanks.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.

Fill in the blanks. Horizontal shifts, vertical shifts, and reflections are called ________ transformations.

The simplest mathematical model for relating two variables is the _______ equation in two variables \(y=m x+b\).

An ordered pair \((a, b)\) is a _________ of an equation in \(x\) and \(y\) when the substitutions \(x=a\) and \(y=b\) result in a true statement.

An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ________ plane.

Fill in the blanks.The inverse function of \(f\) is denoted by _______ .

Match each function with its name. (a) squaring function (d) linear function (g) greatest integer function (b) square root function (e) constant function (h) reciprocal function (c) cubic function (f) absolute value function (i) identity function $$f(x)=x$$

Fill in the blanks. 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 \(h(x)\)=_____.

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

Fill in the blanks. 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 \(y\) is the range.

For a line, the ratio of the change in \(y\) to the change in \(x\) is called the_______ of the line

The point of intersection of the-and-axes is the ________, and the two axes divide the coordinate plane into four parts called ________.

Fill in the blanks. The linear model with the least sum of square differences is called the _____ _____ _____ line.

Fill in the blanks.The domain of \(f\) is the_______ of \(f^{-1},\) and the ______ of \(f^{-1}\) is the range of \(f\).

Match each function with its name. (a) squaring function (d) linear function (g) greatest integer function (b) square root function (e) constant function (h) reciprocal function (c) cubic function (f) absolute value function (i) identity function $$f(x)=1 / x$$

Fill in the blanks. A nonrigid transformation of \(y=f(x)\) represented by \(g(x)=c f(x)\) is a _____ _____ when \(c > 1\) and a _____ _____ when \(0< c <1.\)

A function \(f\) is ______ on an interval when, for any \(x_{1}\) and \(x_{2}\) in the interval, \(x_{1}f\left(x_{2}\right).\)

Fill in the blanks. 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_______ , ______ - form of the equation of a line with slope \(m\) passing through \(\left(x_{1}, y_{1}\right)\) is \(y-y_{1}=m\left(x-x_{1}\right)\).

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

The ________ ________ is a result derived from the Pythagorean Theorem.

Fill in the blanks. An \(r\) -value of a set of data, also called a _____ _____, gives a measure of how well a model fits a set of data.

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

Match each function with its name. (a) squaring function (d) linear function (g) greatest integer function (b) square root function (e) constant function (h) reciprocal function (c) cubic function (f) absolute value function (i) identity function $$f(x)=x^{2}$$

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

A function value \(f(a)\) is a relative _______ of \(f\) when there exists an interval \(\left(x_{1}, x_{2}\right)\) containing \(a\) such that \(x_{1}

Fill in the blanks. In calculus, one of the basic definitions is that of a ____________ ___________ given by \(\frac{f(x+h)-f(x)}{h}, \quad h \neq 0\).

Two lines are ________ if and only if their slopes are equal.

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

Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the ________ ________.

Fill in the blanks. Direct variation models can be described as " \(y\) varies directly as \(x, "\) or " \(y\) is ____ _____ to \(x\) ".

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

Find \((a)(f+g)(x),(b)(f-g)(x)\) (c) \((f g)(x),\) and \((d)(f g)(x) .\) What is the domain of \(f g ?\) $$f(x)=x+2, \quad g(x)=x-2$$

Match each function with its name. (a) squaring function (d) linear function (g) greatest integer function (b) square root function (e) constant function (h) reciprocal function (c) cubic function (f) absolute value function (i) identity function $$f(x)=\sqrt{x}$$

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) \(f(x)=|x-c|\)

Two lines are __________ if and only if their slopes are negative reciprocals of each other.

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

Plot the points in the Cartesian plane. $$(-4,2),(-3,-6),(0,5),(1,-4),(0,0),(3,1)$$

Fill in the blanks.A graphical test for the existence of an inverse function of \(f\) is called the ________ Line Test.

Find \((a)(f+g)(x),(b)(f-g)(x)\) (c) \((f g)(x),\) and \((d)(f g)(x) .\) What is the domain of \(f g ?\) $$f(x)=2 x-5, \quad g(x)=2-x$$

A function \(f\) is ____ when, for each \(x\) in the domain of \(f, f(-x)=-f(x).\)

When the \(x\) -axis and \(y\) -axis have different units of measure, the slope can be interpreted as a _______________ .

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

Plot the points in the Cartesian plane. $$\left(1,-\frac{1}{3}\right),(0.5,-1),\left(\frac{3}{7}, 3\right),\left(-\frac{4}{3},-\frac{3}{7}\right),(-2,2.5)$$

Fill in the blanks. The direct variation model \(y=k x^{n}\) can be described as "y varies directly as the \(n\) th power of \(x\)," or "y is ____ ____ to the \(n\) th power of \(x .\)

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=6 x$$

Find \((a)(f+g)(x),(b)(f-g)(x)\) (c) \((f g)(x),\) and \((d)(f g)(x) .\) What is the domain of \(f g ?\) $$f(x)=x^{2}, \quad g(x)=4 x-5$$

Match each function with its name. (a) squaring function (d) linear function (g) greatest integer function (b) square root function (e) constant function (h) reciprocal function (c) cubic function (f) absolute value function (i) identity function $$f(x)=|x|$$

Determine whether the relation represents \(y\) as a function of \(x\). $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input, } x & 10 & 7 & 4 & 7 & 10 \\ \hline \text { Output, } y & 3 & 6 & 9 & 12 & 15 \\ \hline \end{array}$$