- Last updated

- Save as PDF

- Page ID
- 153748

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vectorC}[1]{\textbf{#1}}\)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

##### Definitions

**function:**a rule that assigns to each element \(x\) of a set \(A\) exactly one element, \(f(x)\), in a set \(B\). Aka, a function takes each input \((x-)\)value and gives a single output \( (y-)\)value.**domain:**the set \(A\) of possible (legal) inputs for a particular function.**range:**the set of output values that a function actually attains.**piecewise-defined function:**a function whose definition is split into cases for certain inputs or ranges of inputs, written with a large brace on the left.**evaluating****:**evaluating a function at an input value or expression is plugging in that value or expression everywhere the variable appeared in the function definition.**graph:**the graph of a function is the set of all points \( (x,f(x))\) plotted on the coordinate plane. Aka, the \(x\)-value tells you the input, and the function output \(f(x)\) tells you the \(y\)-value that goes with it.**even function:**a function such that plugging in \((-x)\) for \(x\) causes no change to the function after simplification, so \(f(-x) = f(x)\). This looks like being symmetric across the \(y\)-axis (like a butterfly).**odd function:**a function such that plugging in \( (-x)\) for \(x\) makes the same function, but negative, after simplification, so \( f(-x) = - f(x)\). This looks like being symmetric about the origin (180-degree rotation has no effect).**floor function:**a function that rounds down any input to the nearest lower integer. So \( \lfloor 3 \rfloor = 3 \) and \( \lfloor 3.7 \rfloor = 3 \), etc.**composition of functions:**plugging a whole function in as the input to another function. Written \( (f\circ g)(x) = f(g(x)) \).**one-to-one function:**a function that sends distinct inputs to distinct outputs. Aka, it never sends multiple inputs to the same output. Example: \(f(x) = x^2 \) is not one-to-one because both \(-2, 2\) are sent to \(4\).**inverse functions:**functions that undo each other, or functions such that if you do one and then the other, you get back the original input. The inverse of \(f\) is \(f^{-1}\), with opposite domain and range, such that \( (f \circ f^{-1})(x) = f( f^{-1}(x))=x \) and \( (f^{-1} \circ f)(x) = f^{-1}(f(x))= x \). Example: \( f(x) = x^3 \) and \( f^{-1}(x) = \sqrt[3]{x} \).

##### Evaluating a Function Example

Evaluate \(f(x) = x^2 - x + 1 \) at \(x = 3, \pi, \) and \( x+h \).

- \( f(3) = 3^2 - 3+ 1 = 7\)
- \( f(\pi) = \pi^2 - \pi + 1 \)
- \( f(x+h) = (x+h)^2 - (x+h) + 1 \)

**ALERT:**use parentheses and be careful with subtraction and powers of negative inputs.

##### Vertical Line Test (VLT)

If there is any place on a given graph where you could draw a**vertical line**that**intersects**the graph at**multiple distinct points**, then the graph**cannot**be the graph of a function.

##### Assessing Domain & Range

- Determine domain of a function by figuring out what numbers are legal to plug in for the variable. Look for inputs that cause dividing by zero or taking an even root of negative numbers. Don't allow those inputs into the domain.
- Determine range of a function by figuring out what kind of numbers it can produce. Look for things like even powers or absolute value bars, which can never produce negative numbers, or radical signs, which by convention give the positive roots.

##### Transformations Table

Transformation Description | \(f (x) \rightarrow \) New Function | English Translation | Example | Graph |

Vertical shift up by \(k\) | \( f(x) + k\) | An original function gets a constant added on to the very end, which causes a vertical change. | \(f(x) = x^2,\rightarrow\) \( g(x) = x^2 + 1 \) | |

Vertical shift down by \(k\) | \( f(x) - k \) | An original function gets a constant subtracted onthe very end, which causes a vertical change. | \(f(x) = x^2, \rightarrow\) \( g(x) = x^2 - 1 \) | |

Horizontal shift right by \(h\) | \( f(x-h) \) | In an original function, each instance of the variable was replaced by the variable minus a constant, causing a horizontal change. | \( f(x) = x^2, \rightarrow\) \( g(x) = (x-1)^2 \) | |

Horizontal shiftleftby \( h\) | \( f(x+h) \) | In an original function, each instance of the variable was replaced by the variable plus a constant, causing a horizontal change. | \( f(x) = x^2, \rightarrow\) \(g(x) = (x+1)^2 \) | |

Reflection across the \(x\)-axis | \( - f(x) \) | Flip the sign on the whole original function, which has the effect of flipping the sign on all \(y\)-values. | \( f(x) = x^2, \rightarrow\) \( g(x) = - x^2 \) | |

Reflection across the \( y\)-axis | \( f( -x)\) | In an original function, each instance of the variable \(x\)was replaced by \( (-x)\). | \( f(x) = x^3, \rightarrow\) \( g(x) = (-x)^3 \) | |

Vertical stretch by a factor of \(k\) | \( k \cdot f(x),\) where \( k > 1\) | An entire original function is multiplied by a constant larger than 1, causing a vertical change. | \( f(x) = x^3-x, \rightarrow\) \(g(x) = 3(x^3-x) \) | |

Vertical shrink by a factor of \(k\) | \( k \cdot f(x)\), where \( 0 < k < 1 \) | An entire original function is multiplied by a positive fraction smallerthan 1, causing a vertical change. | \(f(x) = x^3-x, \rightarrow \) \(g(x) = \frac{1}{2}(x^3-x) \) | |

Horizontalstretchby a factor of \(\frac{1}{h}\) | \( f( hx) \), where \(0 < h < 1 \) | In an original function, each instance of \(x\) is replaced by\(x\) times a small positive fraction, causing a horizontal change. | \( f(x) = x^3-x, \rightarrow\) \( g(x) = (\frac{1}{2}x)^3 - \frac{1}{2}x \) | |

Horizontalshrinkby a factor of \(\frac{1}{h}\) | \( f(hx)\), where \( h > 1 \) | In an original function, each instance of \(x\) is replaced by \(x\) times a constant larger than 1, causing a horizontal change. | \( f(x) = x^3 - x, \rightarrow\) \( g(x) = (2x)^3-2x \) |

##### Identifying Transformations

- Look at the operations that were done to the input variable,
**in order.** - For each operation type, note the transformation effect from the table above.
- Write down in words the effects, in order.

Example: \( f(x) = (x-1)^2 + 3 \) could have started out as \(x^2\), then replaced the input with \((x-1)\), which causes a shift right according to the table. Then a \(+3\) was tacked onto the end, which causes a shift up according to the table. So the transformations were: "Shift right by 1 and shift up by 3."

##### Even and Odd Functions

- A function \(f\) is called an
**even function**if \( f(-x) = f(x) \). Aka, reflecting its graph across the \(y\)-axis changes nothing. Aka aka, its graph is symmetric about the \(y\)-axis. - A function \(f\) is called an
**odd function**if \( f(-x) = -f(x) \). Aka, reflecting its graph across the \(y\)-axis has the same effect as reflecting it across the \(x\)-axis. Aka aka, the graph is symmetric about the origin. - A function need not be one or the other! A function can be
**neither**even nor odd.

##### Algebra With Functions

Given two functions, \(f\) and \(g\), you can make:

- their
**sum,**a new function named "\(f+g\)," defined by \( (f+g)(x) = f(x) + g(x) \). - their
**difference,**a new function named "\( f-g \)," defined by \( (f-g)(x) = f(x) - g(x) \). (ALERT: subtract the WHOLE function \(g\), using parentheses!) - their
**product,**a new function named "\( fg\)," defined by \( (fg)(x) = f(x)g(x) \). (ALERT: if the functions have multiple terms, this would require expanding! Again, use parentheses!) - their
**quotient,**a new function named "\( \frac{f}{g} \)," defined by \( \left( \frac{f}{g} \right) (x) = \dfrac{ f(x)}{g(x)} \). (ALERT: any \(x\) such that \(g(x) = 0\) is not allowed in the domain of this new function!)

These functions are defined on domains where both \(f\) and \(g\) are defined, so we say the domain is the intersection of the domains of \(f\) and \(g\). The domain of the quotient of the functions is further restricted because you can't divide by zero.

##### Composing and Decomposing

**Composing **functions: to compose \( (f \circ g)(x) \),

- Translate to nested parentheses \( f(g(x)) \).
- Replace \(g(x) \) with its definition.
- Go through \(f\) and everywhere you see an \(x\), replace it with the whole input expression in parentheses.

Example: for \( f(x) = x^2 + 3 \), \(g(x) = x-1 \),we have \( (f \circ g)(x) = f(g(x)) = f(x-1) = (x-1)^2+3 \).

**Decomposing**: to find \(f\) and \(g\) such that a function \(h = f \circ g\),

- Identify what happened to the input and in what order. Write in words if needed.
- Split the operations at an "and then" phrase. Use the first activities as the inside function \( g\) and the later activities as \(f\).
- Check answer by composing to see if it matches \(h\).

Example: \( h(x) = \sqrt{ x+2} \) is "add 2 to the input, and then take the square root." Then \( g(x) = x+2\) and \( f(x) = \sqrt{x} \).

##### Horizontal Line Test (HLT)

If there is **any place **on the graph of a function where you can draw a** horizontal line intersecting the graph multiple times,** then it is**not**one-to-one.

##### Finding the Inverse of a Function

To find the inverse of a one-to-one function \(f (x)\),

- Replace "\(f(x)\)" with "\(y\)."
- Solve this equation for \(x\) in terms of \(y\), if possible.
- Swap the variable names: replace all the \(x\)'s with \(y\)'s and vice versa.
- Replace \(y\) with "\(f^{-1}(x)\)."

Note: in an application problem, the variable names are usually descriptive in some way, so skip Step 3 in those.