To solve the problem, we need to find [tex]\( h(10) - g(10) \)[/tex] using the functions provided:
1. Define the functions:
[tex]\( h(a) = 3a + 1 \)[/tex]
[tex]\( g(a) = -a^2 + 1 \)[/tex]
2. Calculate [tex]\( h(10) \)[/tex]:
Substitute [tex]\( a = 10 \)[/tex] into the function [tex]\( h(a) \)[/tex]:
[tex]\[
h(10) = 3 \cdot 10 + 1
\][/tex]
[tex]\[
h(10) = 30 + 1
\][/tex]
[tex]\[
h(10) = 31
\][/tex]
3. Calculate [tex]\( g(10) \)[/tex]:
Substitute [tex]\( a = 10 \)[/tex] into the function [tex]\( g(a) \)[/tex]:
[tex]\[
g(10) = -(10)^2 + 1
\][/tex]
[tex]\[
g(10) = -100 + 1
\][/tex]
[tex]\[
g(10) = -99
\][/tex]
4. Find [tex]\( h(10) - g(10) \)[/tex]:
[tex]\[
h(10) - g(10) = 31 - (-99)
\][/tex]
When subtracting a negative number, it is equivalent to adding the absolute value of that number:
[tex]\[
h(10) - g(10) = 31 + 99
\][/tex]
[tex]\[
h(10) - g(10) = 130
\][/tex]
Hence, the values are:
[tex]\[
h(10) = 31
\][/tex]
[tex]\[
g(10) = -99
\][/tex]
And the required result is:
[tex]\[
h(10) - g(10) = 130
\][/tex]
