Answered

Let's work through another example together.

Problem

Given [tex]f(x) = x^2 + 3x - 3[/tex] and [tex]g(x) = 2x[/tex], find [tex]f(g(x))[/tex].

Solution

First, we need to decide which function will go inside which. Since our function reads " [tex]f[/tex] of [tex]g[/tex] of [tex]x[/tex] ", we know that we are going to substitute the function for [tex]g(x)[/tex] into every [tex]x[/tex] spot in the [tex]f(x)[/tex] function.

[tex]\[ f(g(x)) = (2x)^2 + 3(2x) - 3 \][/tex]

Next, perform normal order of operations for each term. (Hint: [tex]\((2x)^2 = (2x)(2x)\)[/tex])

[tex]\[ f(g(x)) = 4x^2 + 6x - 3 \][/tex]

Let's look at how this example would be different if it was performed in the opposite order.

Given [tex]f(x) = x^2 + 3x - 3[/tex] and [tex]g(x) = 2x[/tex], find [tex]g(f(x))[/tex].

In this case, we'll need to plug the whole function for [tex]f(x)[/tex] into the input [tex]x[/tex] spot in the [tex]g(x)[/tex] function.

[tex]\[ g(f(x)) = 2 \left( x^2 + 3x - 3 \right) \][/tex]

Then we'll distribute to each term.

[tex]\[ g(f(x)) = 2(x^2) + 2(3x) - 2(3) \][/tex]

Simplify to get the final answer.

[tex]\[ g(f(x)) = 2x^2 + 6x - 6 \][/tex]



Answer :

GinnyAnswer
Sure, let's work through the problem step-by-step.

Problem:
Given [tex]\( f(x)=x^2 + 3x - 3 \)[/tex] and [tex]\( g(x)=2x \)[/tex], find [tex]\( f(g(x)) \)[/tex].

Solution:
First, we need to decide which function will go inside which. Since our task is to find " [tex]\( f \)[/tex] of [tex]\( g \)[/tex] of [tex]\( x \)[/tex] ", it means we are going to substitute the function for [tex]\( g(x) \)[/tex] into every [tex]\( x \)[/tex] spot in the [tex]\( f(x) \)[/tex] function.

So, let's substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(2x) \][/tex]

Now, substitute [tex]\( 2x \)[/tex] (which is [tex]\( g(x) \)[/tex]) into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(2x) = (2x)^2 + 3(2x) - 3 \][/tex]

Next, perform the operations for each term.
- First term: [tex]\((2x)^2 = 4x^2\)[/tex]
- Second term: [tex]\(3(2x) = 6x\)[/tex]

So we have:
[tex]\[ f(g(x)) = 4x^2 + 6x - 3 \][/tex]

Let's confirm the given solution using a specific [tex]\( x \)[/tex] value. For example, if [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 2(1) = 2 \][/tex]
Then:
[tex]\[ f(g(1)) = f(2) = (2)^2 + 3(2) - 3 = 4 + 6 - 3 = 7 \][/tex]

The calculated value in this specific example where [tex]\( x = 1 \)[/tex] indeed equals 7, confirming the correctness of our functions and substitution.

### Additional Example: [tex]\( g(f(x)) \)[/tex]
Let's also consider how this example would be different if we wanted to find [tex]\( g(f(x)) \)[/tex].

Given:
[tex]\[ f(x) = x^2 + 3x - 3 \][/tex]
[tex]\[ g(x) = 2x \][/tex]

For [tex]\( g(f(x)) \)[/tex], we need to plug the whole function [tex]\( f(x) \)[/tex] into the input [tex]\( x \)[/tex] spot in the [tex]\( g(x) \)[/tex] function.

Here's the process:
[tex]\[ g(f(x)) = g(x^2 + 3x - 3) \][/tex]

Since [tex]\( g(x) = 2x \)[/tex], we substitute [tex]\( x^2 + 3x - 3 \)[/tex] into [tex]\( x \)[/tex]:
[tex]\[ g(f(x)) = 2(x^2 + 3x - 3) \][/tex]

Then we distribute [tex]\( 2 \)[/tex] to each term inside the parentheses:
[tex]\[ g(f(x)) = 2(x^2) + 2(3x) - 2(3) \][/tex]

Simplify each term:
[tex]\[ g(f(x)) = 2x^2 + 6x - 6 \][/tex]

So the final expression for [tex]\( g(f(x)) \)[/tex] is:
[tex]\[ g(f(x)) = 2x^2 + 6x - 6 \][/tex]

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