Math Problem Statement

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)?

Solution

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

f(x)=(10x2+7x+4)mod12f(x) = (10x^2 + 7x + 4) \mod 12

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

Step 1: Compute f(0)f(0)

For x=0x = 0: f(0)=(10(0)2+7(0)+4)mod12=4mod12=4f(0) = (10(0)^2 + 7(0) + 4) \mod 12 = 4 \mod 12 = 4

Step 2: Compute f(7)f(7)

For x=7x = 7: f(7)=(10(7)2+7(7)+4)mod12=(10×49+7×7+4)mod12f(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: 10×49=490,7×7=49,so f(7)=(490+49+4)mod12=543mod1210 \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: 543mod12=3543 \mod 12 = 3

Step 3: Combine results

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

So, the result is: f(T)={3,4}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)f(3) for x=3x = 3?
  2. How would the set f(T)f(T) change if T={1,2}T = \{1, 2\}?
  3. Can you generalize f(x)f(x) for any xSx \in S modulo 12?
  4. How does the choice of modulo (e.g., mod 12 vs. mod 5) affect the results of f(x)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.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Quadratic Functions

Formulas

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

Theorems

Modulo Operation
Polynomial Function Evaluation

Suitable Grade Level

Grades 9-12