Objective 3 - One-to-One

Is the following graph 1-1?

$\graph [rectangular]{f(x) = 3x+4}$

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

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following graph 1-1?

$\graph [rectangular]{f(x) = -(x-2)^2+3}$

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

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

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following graph 1-1?

$\graph [rectangular]{f(x) = x^3 - 1}$

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

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following graph 1-1?

$\graph [rectangular]{f(x) = (x-1)^{1/2}+3}$

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

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

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following graph 1-1?

$\graph [rectangular]{f(x) = (x+2)^{1/3}-1}$

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

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

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following graph 1-1?

$\graph [rectangular]{f(x) = 1/(x+1)}$

If you are having trouble seeing the Desmos graph, use the “zoom out” button. You can also interact with the graph! Click and move to look around.

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

”Yes” or ”No”

To check if an equation is a function, we have the [blank] Line Test. Do we have a similar way to test if a function is 1-1?

Is the following function 1-1?

$f(x) = 3x+4$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

Is the following function 1-1?

$f(x) = -(x-2)^2+3$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

Is the following function 1-1?

$f(x) = x^3 - 1$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

Is the following function 1-1?

$f(x) = \sqrt {x-1}+3$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

Is the following function 1-1?

$f(x) = \sqrt [3]{x+2}-1$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

Is the following function 1-1?

$f(x) = \frac {1}{x+1}$

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

”Yes” or ”No”

To algebraically check if an equation is a function, we check if $f(a) = f(b)$ leads to $a = b$ . That is, you plug in $a$ , plug in $b$ , then set them equal to each other and try to reduce. If only there were an easier way to check...

When you started this Objective, you were likely put off by the number of questions. This was done purposefully! Before moving on with the summary, have you finished all of the previous questions?

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

If you look back, the first 6 questions are exactly the same as questions 7 through 12! This was done to illustrate that knowing the general shapes of functions can give you useful information! In fact, this section can be done very quickly when using the Vertical Line Test. As you learn about new functions and new Coodinate Systems, the Vertical Line Test will stop telling you whether a function is 1-1. But we don’t have to worry about that now.

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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