Math  /  Algebra

QuestionThe function ff is defined by f(x)=x23f(x)=x^{2}-3. Find f(2x)f(2 x). f(2x)=f(2 x)=

Studdy Solution

STEP 1

What is this asking? Given a function f(x)f(x) that involves squaring and subtracting, we need to find the value of the function when its input is doubled. Watch out! Don't forget to square the *entire* input (2x)(2x), not just the xx!

STEP 2

1. Define the function
2. Substitute the new input
3. Simplify the expression

STEP 3

We're given the function f(x)=x23f(x) = x^2 - 3.
This tells us that whatever we put into the function, we **square it** and then **subtract 3**.
It's like a little machine that takes a number, processes it, and spits out a new one!

STEP 4

Now, instead of just xx, we're asked to find f(2x)f(2x).
This means our input is now **double** the original xx.
So, let's **replace** every xx in our original function with (2x)(2x).

STEP 5

f(2x)=(2x)23f(2x) = (2x)^2 - 3 Notice how the entire (2x)(2x) is being squared!
This is super important!

STEP 6

Let's **expand** the squared term: (2x)2(2x)^2 means (2x)(2x)(2x) \cdot (2x).

STEP 7

(2x)(2x)=2x2x=22xx=4x2=4x2(2x) \cdot (2x) = 2 \cdot x \cdot 2 \cdot x = 2 \cdot 2 \cdot x \cdot x = 4 \cdot x^2 = 4x^2

STEP 8

So, we have: f(2x)=4x23f(2x) = 4x^2 - 3 And that's our **simplified expression**!

STEP 9

f(2x)=4x23f(2x) = 4x^2 - 3

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