Objective 4 - Inverse

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Find the inverse of a function, if it exists.

Link to section in online textbook

First, watch this video to learn when a function has an inverse and how to find the inverse of a function. Feel free to pause the video and fill out the notes as you go.

Determine whether the function below is 1-1.

$f(x) = \sage {f1}$

$\answer [format=string]{Yes}$

“Yes” or “No”.

If $f(x)$ is 1-1, find the inverse and define the domain on which the inverse is valid. If $f(x)$ is not 1-1, put “NA” for all answer blocks.

$f^{-1}(x) = \answer {\sage {f1Inverse}}$

To find the inverse of a function, switch $x$ and $y$ , then solve for $y$ . Don’t round.

Domain of $f^{-1}(x)$ : $\answer [format=string]{(} \answer {\sage {-Infinity}}, \answer {\sage {Infinity}} \answer [format=string]{)}$

Think about the shape of the original function: are there places whether the function is not defined?

Determine whether the function below is 1-1.

$f(x) = \sage {f2}$

$\answer [format=string]{No}$

“Yes” or “No”.

If $f(x)$ is 1-1, find the inverse and define the domain on which the inverse is valid. If $f(x)$ is not 1-1, put “NA” for all answer blocks.

$f^{-1}(x) = \answer [format=string]{NA}$

To find the inverse of a function, switch $x$ and $y$ , then solve for $y$ . Don’t round.

Domain of $f^{-1}(x)$ : $\answer [format=string]{NA} \answer [format=string]{NA}, \answer [format=string]{NA} \answer [format=string]{NA}$

Think about the shape of the original function: are there places whether the function is not defined?

Determine whether the function below is 1-1.

$f(x) = \sage {f3}$

$\answer [format=string]{Yes}$

“Yes” or “No”.

If $f(x)$ is 1-1, find the inverse and define the domain on which the inverse is valid. If $f(x)$ is not 1-1, put “NA” for all answer blocks.

$f^{-1}(x) = \answer {\sage {f3Inverse}}$

To find the inverse of a function, switch $x$ and $y$ , then solve for $y$ . Don’t round.

Domain of $f^{-1}(x)$ : $\answer [format=string]{(} \answer {\sage {-Infinity}}, \answer {\sage {Infinity}} \answer [format=string]{)}$

Think about the shape of the original function: are there places whether the function is not defined?

Determine whether the function below is 1-1.

$f(x) = \sage {f4}$

$\answer [format=string]{Yes}$

“Yes” or “No”.

If $f(x)$ is 1-1, find the inverse and define the domain on which the inverse is valid. If $f(x)$ is not 1-1, put “NA” for all answer blocks.

$f^{-1}(x) = \answer {\sage {f4Inverse}}$

To find the inverse of a function, switch $x$ and $y$ , then solve for $y$ . Don’t round.

Domain of $f^{-1}(x)$ : $\answer [format=string]{(} \answer {\sage {-Infinity}}, \answer {\sage {domain4[1]}} \answer [format=string]{]}$

Think about the shape of the original function: are there places whether the function is not defined?

Determine whether the function below is 1-1.

$f(x) = \sage {f5}$

$\answer [format=string]{Yes}$

“Yes” or “No”.

If $f(x)$ is 1-1, find the inverse and define the domain on which the inverse is valid. If $f(x)$ is not 1-1, put “NA” for all answer blocks.

$f^{-1}(x) = \answer {\sage {f5Inverse}}$

To find the inverse of a function, switch $x$ and $y$ , then solve for $y$ . Don’t round.

Domain of $f^{-1}(x)$ : $\answer [format=string]{(} \answer {\sage {-Infinity}}, \answer {\sage {Infinity}} \answer [format=string]{)}$

Think about the shape of the original function: are there places whether the function is not defined?

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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