Suppose 𝑆={0,1,2,3,4,5,6,7,8,9,10,11}S={0,1,2,3,4,5,6,7,8,9,10,11} and that the function 𝑓:𝑆→𝑆f:S→S is given by:

𝑓(𝑥)=(10𝑥2+7𝑥+4)mod12f(x)=(10x2+7x+4)mod12.

Let 𝑇={0,7}T={0,7}.

Recall that the Numbas syntax for the set {𝑎,𝑏,𝑐}{a,b,c} is `set(a,b,c)`.

a)

What is 𝑓(𝑇)f(T)?

We are given the function \( f: S \to S \) defined as:

\[
f(x) = (10x^2 + 7x + 4) \mod 12
\]

We are also given the set \( T = \{0, 7\} \). Our goal is to compute \( f(T) \), which means finding the values of \( f(x) \) for \( x \in T \).

### Step 1: Compute \( f(0) \)
For \( x = 0 \):
\[
f(0) = (10(0)^2 + 7(0) + 4) \mod 12 = 4 \mod 12 = 4
\]

### Step 2: Compute \( f(7) \)
For \( x = 7 \):
\[
f(7) = (10(7)^2 + 7(7) + 4) \mod 12 = (10 \times 49 + 7 \times 7 + 4) \mod 12
\]
First, compute the values inside the parentheses:
\[
10 \times 49 = 490,\quad 7 \times 7 = 49,\quad \text{so } f(7) = (490 + 49 + 4) \mod 12 = 543 \mod 12
\]
Now compute the modulo operation:
\[
543 \mod 12 = 3
\]

### Step 3: Combine results
Now that we have \( f(0) = 4 \) and \( f(7) = 3 \), we can write the set \( f(T) \) as:
\[
f(T) = \{3, 4\}
\]

So, the result is:
\[
f(T) = \{3, 4\}
\]

Do you want more details or have any questions?

Here are 5 related questions you can consider:
1. What is \( f(3) \) for \( x = 3 \)?
2. How would the set \( f(T) \) change if \( T = \{1, 2\} \)?
3. Can you generalize \( f(x) \) for any \( x \in S \) modulo 12?
4. How does the choice of modulo (e.g., mod 12 vs. mod 5) affect the results of \( f(x) \)?
5. What if the polynomial were cubic instead of quadratic?

**Tip:** When computing functions with modular arithmetic, it's often easier to simplify intermediate steps modulo the given number.

Suppose S = {0,1,2,3,4,5,6,7,8,9,10,11} and that the function f: S → S is given by: f(x) = (10x^2 + 7x + 4) mod 12. Let T = {0, 7}. What is f(T)?

Explore how to evaluate the function f(x) = (10x^2 + 7x + 4) mod 12 for the set T = {0, 7}. This problem involves key mathematical concepts like modular arithmetic and quadratic functions, suitable for high school students (Grades 9-12).

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