Composition and Inverse
Name
MAT222
Janice Reyes
May 26, 2014
Composition and Inverse
The following functions will be used in this week’s assignment.
1.fx=2x+5
2. gx=x2-3
3. hx=7-x3
f-h(4)- Original expression
f4-h4- Rewritten
24+5-7-43 - Substitute 4 for x
8+5- 33 - Simplify
13-1- Simplify further
12-Solution
The second problem involves composing two pairs into each other.
f°g(x)- Original expression
f(gx)- Rewritten so that rule of f will apply
fx2-3- g inserted
2x2-3+5- f applied to g
2x2-6+5- Simplified
2x2-1- Solution
The third problem also involves composing two pairs into each other.
h°g(x)- Original expression
h(g(x))- Rewritten so that rule of h will apply to g
hx2-3- g inserted
7-(x2-3)3- h applied to g
21-3(x2-3)- Multiply numerator and denominator by 3
21-3x2-9- Simplified
12-3x2- Simplified
-3x2+12- Rewritten
x2+4- Solution
The fourth problem involves transforming the function gx=(x2-3) so that it appears on a different x and y axis. In order to transform the g(x) function six points to the right and seven down, we must modify the equation in the following way: gx=(x2-3) - Original function
gx=((x-6)2-3) - Subtracting six from x will transform the function six spaces to the right gx=x-62-3-7 - Subtracting seven from the entire function will transform the function seven spaces down Lastly, we must find the inverse of both f and h. To do so, we will simply rewrite the function with y instead of the function name, switch y and x and then solve for y. Then we will rename y again to the inverse function name to solve. To solve: f-1(x)
fx=2x+5- Original function
y=2x+5 - Rewritten with y replacing function name
x=2y+5 - Rewritten with x and y transposed
2y=x-5 - Solving for y
y=x-52- Solved for y
f-1x=x-52 - Solved.
To solve h-1(x):
hx=7-x3 - Original function
y=7-x3 - Rewritten with y replacing function name
x=7-y3 - Rewritten with x and y transposed
3x=7-y - Multiply both sides by 3
-y=3x-7 - Isolate y...