Answered

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If [tex][tex]$f(x)=x^3-2x^2+3x-5$[/tex][/tex] and [tex][tex]$g(x)=x^2+x-1$[/tex][/tex], perform the given function operation.

Find [tex][tex]$f(g(x))$[/tex][/tex]. Do not expand or simplify.



Answer :

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]