Question

use these functions to answer #3-13. show your work!
$f(x) = x - 3$
$g(x) = x^2 + 9x + 3$
$h(x) = 2x + 5$
hint: #3-5 should be numerical answers!

1. $f(h(6)) = \underline{\\ 14\\ }$

$h(6)=2(6)+5=17$
$f(17)=17 - 3 = 14$

1. $(h \circ g)(3) = \underline{\\ 83\\ }$

$g(3)=3^2 + 9(3) + 3 = 9 + 27 + 3 = 39$

1. $(g \circ f)(7) = \underline{\\ 55\\ }$

$f(7)=7 - 3 = 4$
$g(4)=4^2 + 9(4) + 3 = 16 + 36 + 3 = $

1. write a new function to represent $(f \circ h)(x)$.
2. write a new function to represent $h(g(x))$.
3. $h(x) + g(x)$
4. $g(x) - f(x)$
5. $g(x) - f(x) + h(x)$
6. $2(f(x)) + 3(h(x))$
7. $h(x) \cdot f(x)$
8. $g(x) \cdot h(x)$

Explanation:

Step1: Solve $(f \circ h)(x)$

Substitute $h(x)$ into $f(x)$:
$f(h(x)) = (2x + 5) - 3$
Simplify: $f(h(x)) = 2x + 2$

Step2: Solve $h(g(x))$

Substitute $g(x)$ into $h(x)$:
$h(g(x)) = 2(x^2 + 9x + 3) + 5$
Expand and simplify: $h(g(x)) = 2x^2 + 18x + 6 + 5 = 2x^2 + 18x + 11$

Step3: Solve $h(x) + g(x)$

Add the two functions:
$h(x)+g(x) = (2x + 5) + (x^2 + 9x + 3)$
Combine like terms: $h(x)+g(x) = x^2 + 11x + 8$

Step4: Solve $g(x) - f(x)$

Subtract $f(x)$ from $g(x)$:
$g(x)-f(x) = (x^2 + 9x + 3) - (x - 3)$
Expand and simplify: $g(x)-f(x) = x^2 + 9x + 3 - x + 3 = x^2 + 8x + 6$

Step5: Solve $g(x) - [f(x) + h(x)]$

First find $f(x)+h(x)$:
$f(x)+h(x) = (x - 3) + (2x + 5) = 3x + 2$
Subtract from $g(x)$:
$g(x)-[f(x)+h(x)] = (x^2 + 9x + 3) - (3x + 2)$
Simplify: $g(x)-[f(x)+h(x)] = x^2 + 6x + 1$

Step6: Solve $2(f(x)) + 3(h(x))$

Multiply each function then add:
$2(f(x))+3(h(x)) = 2(x - 3) + 3(2x + 5)$
Expand and simplify: $2(f(x))+3(h(x)) = 2x - 6 + 6x + 15 = 8x + 9$

Step7: Solve $h(x) \cdot f(x)$

Multiply the two functions:
$h(x)\cdot f(x) = (2x + 5)(x - 3)$
Expand using FOIL: $h(x)\cdot f(x) = 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15$

Step8: Solve $g(x) \cdot h(x)$

Multiply the two functions:
$g(x)\cdot h(x) = (x^2 + 9x + 3)(2x + 5)$
Expand and simplify:
$g(x)\cdot h(x) = 2x^3 + 5x^2 + 18x^2 + 45x + 6x + 15 = 2x^3 + 23x^2 + 51x + 15$

Answer:

1. $\boldsymbol{2x + 2}$
2. $\boldsymbol{2x^2 + 18x + 11}$
3. $\boldsymbol{x^2 + 11x + 8}$
4. $\boldsymbol{x^2 + 8x + 6}$
5. $\boldsymbol{x^2 + 6x + 1}$
6. $\boldsymbol{8x + 9}$
7. $\boldsymbol{2x^2 - x - 15}$
8. $\boldsymbol{2x^3 + 23x^2 + 51x + 15}$