Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$, for some integers $a, b$. Determine $a, b$.

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> [!TLDR]
> Use any two points from the given set to form simultaneous equations in $a$ and $b$; solving reveals the linear function is $f(x) = 2x - 1$, so $a = 2$ and $b = -1$.

A function defined by $f(x) = ax + b$ is a linear function, completely determined by two parameters: the slope $a$ and the $y$-intercept $b$. When we're given specific input-output pairs, we can substitute them into the function's formula to create equations that pin down these unknowns.

The set $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ tells us that $f(1) = 1$, $f(2) = 3$, $f(0) = -1$, and $f(-1) = -3$. Since the function has the form $f(x) = ax + b$, any two of these pairs will suffice to find $a$ and $b$. The remaining pairs then serve as a check.

Let me use the simplest pair first: $(0, -1)$.

1. **Extract $b$ directly from $f(0) = -1$:**
   
   Substituting $x = 0$ into $f(x) = ax + b$ gives:
   $$f(0) = a \cdot 0 + b = b$$
   
   Since $f(0) = -1$, we immediately have $b = -1$.

2. **Find $a$ using another point, say $(1, 1)$:**
   
   Substitute $x = 1$ and $f(1) = 1$ into $f(x) = ax + b$:
   $$1 = a \cdot 1 + b = a + b$$
   
   We already know $b = -1$, so:
   $$1 = a - 1 \implies a = 2$$

3. **Verify with the remaining points:** …

Q.Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$ , for some integers $a, b$ . Determine $a, b$ .

Concept understanding — Linear Function Evaluation

Linear Function Evaluation — From Intuition to Precision

The Intuition: A Machine That Adds the Same Amount Each Time

The Precise Statement

How to Evaluate — Step by Step

Why This Matters for Exams

Q.Let f={(1,1),(2,3),(0,−1),(−1,−3)}f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}f={(1,1),(2,3),(0,−1),(−1,−3)} be a function from Z\mathbb{Z}Z to Z\mathbb{Z}Z defined by f(x)=ax+bf(x) = ax + bf(x)=ax+b, for some integers a,ba, ba,b. Determine a,ba, ba,b.

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Q.Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$ , for some integers $a, b$ . Determine $a, b$ .

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