Answer :
We are given the functions
[tex]$$
f(x)=1.25x,\quad g(x)=0.15625x^2,\quad h(x)=1.31^x.
$$[/tex]
We wish to:
1. Evaluate these functions at [tex]$x=6,\;8,\;12$[/tex].
2. Determine the smallest integer [tex]$x$[/tex] when [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex].
–––––––––––
Step 1. Evaluation for [tex]$x=6,\;8,\;12$[/tex]
–––––––––––
For function [tex]$f(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
f(6)=1.25 \times 6=7.5.
$$[/tex]
• When [tex]$x=8$[/tex]:
[tex]$$
f(8)=1.25 \times 8=10.0.
$$[/tex]
• When [tex]$x=12$[/tex]:
[tex]$$
f(12)=1.25 \times 12=15.0.
$$[/tex]
–––––––––––
For function [tex]$g(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
g(6)=0.15625 \times 6^2=0.15625 \times 36=5.625.
$$[/tex]
• When [tex]$x=8$[/tex]:
[tex]$$
g(8)=0.15625 \times 8^2=0.15625 \times 64=10.0.
$$[/tex]
• When [tex]$x=12$[/tex]:
[tex]$$
g(12)=0.15625 \times 12^2=0.15625 \times 144=22.5.
$$[/tex]
–––––––––––
For function [tex]$h(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
h(6)=1.31^6\approx 5.054,
$$[/tex]
rounded to the nearest thousandth.
• When [tex]$x=8$[/tex]:
[tex]$$
h(8)=1.31^8\approx 8.673,
$$[/tex]
rounded as needed.
• When [tex]$x=12$[/tex]:
[tex]$$
h(12)=1.31^{12}\approx 25.542,
$$[/tex]
rounded to the nearest thousandth.
–––––––––––
Step 2. Finding when [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]
We wish to find the smallest integer [tex]$x$[/tex] such that
[tex]$$
h(x)>f(x)\quad\text{and}\quad h(x)>g(x).
$$[/tex]
Testing at [tex]$x=1$[/tex]:
• Calculate
[tex]$$
f(1)=1.25 \times 1=1.25,
$$[/tex]
[tex]$$
g(1)=0.15625 \times 1^2=0.15625,
$$[/tex]
[tex]$$
h(1)=1.31^1=1.31.
$$[/tex]
Since
[tex]$$
1.31>1.25 \quad \text{and}\quad 1.31>0.15625,
$$[/tex]
both conditions are satisfied at [tex]$x=1$[/tex].
–––––––––––
Conclusion
a) The function evaluations are:
[tex]$$
\begin{array}{llll}
&x=6& x=8 &x=12\\[6pt]
f(x)&7.5&10.0&15.0\\[6pt]
g(x)&5.625&10.0&22.5\\[6pt]
h(x)&\approx 5.054&\approx 8.673&\approx 25.542\\
\end{array}
$$[/tex]
b) The smallest integer [tex]$x$[/tex] for which [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] is [tex]$x=1$[/tex].
[tex]$$
f(x)=1.25x,\quad g(x)=0.15625x^2,\quad h(x)=1.31^x.
$$[/tex]
We wish to:
1. Evaluate these functions at [tex]$x=6,\;8,\;12$[/tex].
2. Determine the smallest integer [tex]$x$[/tex] when [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex].
–––––––––––
Step 1. Evaluation for [tex]$x=6,\;8,\;12$[/tex]
–––––––––––
For function [tex]$f(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
f(6)=1.25 \times 6=7.5.
$$[/tex]
• When [tex]$x=8$[/tex]:
[tex]$$
f(8)=1.25 \times 8=10.0.
$$[/tex]
• When [tex]$x=12$[/tex]:
[tex]$$
f(12)=1.25 \times 12=15.0.
$$[/tex]
–––––––––––
For function [tex]$g(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
g(6)=0.15625 \times 6^2=0.15625 \times 36=5.625.
$$[/tex]
• When [tex]$x=8$[/tex]:
[tex]$$
g(8)=0.15625 \times 8^2=0.15625 \times 64=10.0.
$$[/tex]
• When [tex]$x=12$[/tex]:
[tex]$$
g(12)=0.15625 \times 12^2=0.15625 \times 144=22.5.
$$[/tex]
–––––––––––
For function [tex]$h(x)$[/tex]:
• When [tex]$x=6$[/tex]:
[tex]$$
h(6)=1.31^6\approx 5.054,
$$[/tex]
rounded to the nearest thousandth.
• When [tex]$x=8$[/tex]:
[tex]$$
h(8)=1.31^8\approx 8.673,
$$[/tex]
rounded as needed.
• When [tex]$x=12$[/tex]:
[tex]$$
h(12)=1.31^{12}\approx 25.542,
$$[/tex]
rounded to the nearest thousandth.
–––––––––––
Step 2. Finding when [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]
We wish to find the smallest integer [tex]$x$[/tex] such that
[tex]$$
h(x)>f(x)\quad\text{and}\quad h(x)>g(x).
$$[/tex]
Testing at [tex]$x=1$[/tex]:
• Calculate
[tex]$$
f(1)=1.25 \times 1=1.25,
$$[/tex]
[tex]$$
g(1)=0.15625 \times 1^2=0.15625,
$$[/tex]
[tex]$$
h(1)=1.31^1=1.31.
$$[/tex]
Since
[tex]$$
1.31>1.25 \quad \text{and}\quad 1.31>0.15625,
$$[/tex]
both conditions are satisfied at [tex]$x=1$[/tex].
–––––––––––
Conclusion
a) The function evaluations are:
[tex]$$
\begin{array}{llll}
&x=6& x=8 &x=12\\[6pt]
f(x)&7.5&10.0&15.0\\[6pt]
g(x)&5.625&10.0&22.5\\[6pt]
h(x)&\approx 5.054&\approx 8.673&\approx 25.542\\
\end{array}
$$[/tex]
b) The smallest integer [tex]$x$[/tex] for which [tex]$h(x)$[/tex] exceeds both [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] is [tex]$x=1$[/tex].