Algebra II - Functions and Graphs

Consider the following two functions:

$f(x) = x^{2}$ and $g(x) = (x+3)^{2} - 6$

How is the function shifted compared with ?

units left, units down

units right, units down

units left, units up

units right, units down

units left, units down

Explanation

The $(x+3)^{2}$ portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

Consider the following two functions:

$f(x) = x^{2}$ and $g(x) = (x+3)^{2} - 6$

How is the function shifted compared with ?

units left, units down

units right, units down

units left, units up

units right, units down

units left, units down

Explanation

Determine the inverse:

$y=\frac{1}{10}x-\frac{3}{10}$

$y=\frac{1}{10}x+\frac{3}{10}$

$y=-\frac{1}{10}x-\frac{3}{10}$

$y=\frac{1}{10}x-3$

$y=\frac{3}{10}x-3$

Explanation

In order to find the inverse of this function, interchange the x and y-variables.

Subtract three from both sides.

Simplify the equation.

Divide by ten on both sides.

$\frac{x-3}{10}= \frac{10y}{10}$

Simplify both sides.

The answer is: $y=\frac{1}{10}x-\frac{3}{10}$

Which of the following is NOT a function?

$x=\left | y\right |$

$y = x^{3} -x +5$

$y = \sqrt{x-9} +3$

$y = -\sqrt{x^{2}-16}$

Explanation

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function $x=\left | y\right |$ , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, $y = \sqrt{x-9} +3$ , a cubic equation, $y = x^{3} -x +5$ , and a semicircle, $y = -\sqrt{x^{2}-16}$ . These will all pass the vertical line test.

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The above table shows a function with domain $\left { -3, -2, -1, 0, 1, 2, 3 \right }$ .

True or false: has an inverse function.

False

True

Explanation

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

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and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

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The above table shows a function with domain $\left { -3, -2, -1, 0, 1, 2, 3 \right }$ .

True or false: has an inverse function.

False

True

Explanation

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

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and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

If and , determine: