Certainly! Let's find [tex]\( f(g(x)) \)[/tex] given the functions [tex]\( f(x) = x^3 - 2x^2 + 3x - 5 \)[/tex] and [tex]\( g(x) = x^2 + x - 1 \)[/tex].
### Step-by-Step Solution:
1. Identify the inner function [tex]\( g(x) \)[/tex]:
Let's start with the inner function:
[tex]\[
g(x) = x^2 + x - 1
\][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] with [tex]\( x \)[/tex] replaced by [tex]\( g(x) \)[/tex]:
To find [tex]\( f(g(x)) \)[/tex], we need to substitute [tex]\( g(x) \)[/tex] into the function [tex]\( f(x) \)[/tex].
The function [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = x^3 - 2x^2 + 3x - 5
\][/tex]
We'll replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[
f(g(x)) = (g(x))^3 - 2(g(x))^2 + 3(g(x)) - 5
\][/tex]
3. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
Now replace every [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[
f(g(x)) = (x^2 + x - 1)^3 - 2(x^2 + x - 1)^2 + 3(x^2 + x - 1) - 5
\][/tex]
4. Combine the results:
Putting it all together, we get:
[tex]\[
f(g(x)) = (x^2 + x - 1)^3 - 2(x^2 + x - 1)^2 + 3(x^2 + x - 1) - 5
\][/tex]
There you have it, [tex]\( f(g(x)) \)[/tex] in its composed form without expanding or simplifying further:
[tex]\[
f(g(x)) = (x^2 + x - 1)^3 - 2(x^2 + x - 1)^2 + 3(x^2 + x - 1) - 5
\][/tex]
