Learning how to do these are the last things I need to learn for today.

Graph each function and its inverse.

1) F(x) = |x - 3| + 2

2) F(x) = (x + 2)2 + 3

(The small 2 represents power)

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- Sep 30th 2007, 02:41 PMBrazucaGraph each function and its inverse
Learning how to do these are the last things I need to learn for today.

Graph each function and its inverse.

1) F(x) = |x - 3| + 2

2) F(x) = (x + 2)2 + 3

(The small 2 represents power)

- Sep 30th 2007, 03:03 PMFailbait
http://i20.tinypic.com/o0njic.png

The first function is an inverse of the second and vice versa. Simply replace the $\displaystyle x$'s in your function with $\displaystyle y$'s to receive the inverse function.

$\displaystyle f(x) = (x + 2)^2 + 3 $

$\displaystyle y = (x + 2)^2 + 3$

$\displaystyle x = (y + 2)^2 + 3$

$\displaystyle -(y+2)^2 = -x + 3$

$\displaystyle (y + 2)^2 = x - 3$

$\displaystyle y + 2 = \sqrt{x - 3}$

$\displaystyle y = \sqrt{x - 3} + 2$

Final Inverse of $\displaystyle f(x) = (x + 2)^2 + 3 $:

$\displaystyle f(x) = \sqrt{x - 3} + 2$ - Sep 30th 2007, 05:00 PMBrazuca
- Sep 30th 2007, 05:01 PMFailbait
Ah yes, sorry about that. :)

- Sep 30th 2007, 05:17 PMBrazuca
How would I solve the first problem?

Graph each function and its inverse.

1) F(x) = |x - 3| + 2 - Sep 30th 2007, 05:39 PMJhevon
recall that for absolute values you have to consider two cases, when what's inside absolute value signs is negative and when it is positive.

to find the inverse, we switch x and y and solve for y:

$\displaystyle f(x) = |x - 3| + 2 = \left \{ \begin{array}{cc} x - 1, & \mbox { if } x \ge 3 \\ 5 - x, & \mbox { if } x < 3 \end{array} \right.$

work on each piece separately.

For the first graph:

$\displaystyle y = x - 1$ for $\displaystyle x \ge 3$

For inverse, switch x and y:

$\displaystyle x = y - 1$ for $\displaystyle y \ge 3$

$\displaystyle \Rightarrow y = x + 1$ for $\displaystyle y \ge 3$ (which is equivalent to $\displaystyle x \ge 2$)

now for the second graph:

$\displaystyle y = 5 - x$ for $\displaystyle x < 3$

For inverse, switch x and y:

$\displaystyle x = 5 - y$ for $\displaystyle y < 3$

$\displaystyle \Rightarrow y = 5 - x$ for $\displaystyle y < 3$ (which is equivalent to $\displaystyle x > 2$)

thus, putting these together we have:

$\displaystyle f^{-1}(x) = \left \{ \begin{array}{cc} x + 1, & \mbox { if } x \ge 2 \\ 5 - x, & \mbox { if } x > 2 \end{array} \right.$

the graph is below. the green is $\displaystyle f(x)$, the red is $\displaystyle f^{-1}(x)$