Question

use the functions to answer parts a and b.
$f(x)=0.75x$
$g(x)=0.09375x^{2}$
$h(x)=1.25^{x}$
a. evaluate each function for $x=6$, $x=8$, and $x=12$.
b. when will function h exceed both function f and function g?
a. evaluate $f(x)$ at $x=6$, $x=8$, and $x=12$.
$f(6)=\square$, $f(8)=\square$, and $f(12)=\square$
(type an integer or decimal rounded to the nearest thousandth as needed.)

Explanation:

Step1: Calculate f(6)

Substitute $x=6$ into $f(x)=0.75x$:
$f(6)=0.75\times6=4.5$

Step2: Calculate f(8)

Substitute $x=8$ into $f(x)=0.75x$:
$f(8)=0.75\times8=6$

Step3: Calculate f(12)

Substitute $x=12$ into $f(x)=0.75x$:
$f(12)=0.75\times12=9$

Step4: Calculate g(6)

Substitute $x=6$ into $g(x)=0.09375x^2$:
$g(6)=0.09375\times6^2=0.09375\times36=3.375$

Step5: Calculate g(8)

Substitute $x=8$ into $g(x)=0.09375x^2$:
$g(8)=0.09375\times8^2=0.09375\times64=6$

Step6: Calculate g(12)

Substitute $x=12$ into $g(x)=0.09375x^2$:
$g(12)=0.09375\times12^2=0.09375\times144=13.5$

Step7: Calculate h(6)

Substitute $x=6$ into $h(x)=1.25^x$:
$h(6)=1.25^6\approx3.8147$

Step8: Calculate h(8)

Substitute $x=8$ into $h(x)=1.25^x$:
$h(8)=1.25^8\approx5.9605$

Step9: Calculate h(12)

Substitute $x=12$ into $h(x)=1.25^x$:
$h(12)=1.25^{12}\approx14.5519$

Step10: Find x where h(x) > f(x), h(x) > g(x)

Test values:

• $x=11$: $h(11)=1.25^{11}\approx11.6415$, $f(11)=8.25$, $g(11)=0.09375\times121=11.3438$ → $h(x)$ exceeds both
• $x=10$: $h(10)=1.25^{10}\approx9.3132$, $f(10)=7.5$, $g(10)=0.09375\times100=9.375$ → $h(x) < g(x)$

So $x=11$ is the smallest integer where $h(x)$ exceeds both.

Answer:

Part a:

$f(6)=4.5$, $f(8)=6$, $f(12)=9$
$g(6)=3.375$, $g(8)=6$, $g(12)=13.5$
$h(6)\approx3.815$, $h(8)\approx5.961$, $h(12)\approx14.552$

Part b:

When $x=11$ (the smallest integer value; for real numbers, $x\approx10.9$ so $x>10.9$), $h(x)$ exceeds both $f(x)$ and $g(x)$.