Q.Is $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$ a function? If $g$ is described by $g(x) = \alpha x + \beta$, then what value should be assigned to $\alpha$ and $\beta$?

Concept: Function Equality — two functions are equal if they have the same domain and the same output for every input.

1. The set $g$ is a function because each input (1, 2, 3, 4) maps to exactly one output — no domain element repeats with a different image.

2. We assume $g(x) = \alpha x + \beta$ for all $x$ in $\{1,2,3,4\}$. Using two points:

   From $(1,1)$: $\alpha + \beta = 1$

   From $(2,3)$: $2\alpha + \beta = 3$

3. Subtract the first equation from the second: $\alpha = 2$. Then $\beta = 1 - \alpha = -1$.

4. Check with $(3,5)$: $2(3) - 1 = 5$ ✓ and $(4,7)$: $2(4) - 1 = 7$ ✓ — consistent.

Why this works: the idea of function equality

A function is a rule that assigns each input exactly one output. The set $g$ lists four ordered pairs — each first element (domain value) appears only once, so yes, it’s a function. The question then says: if this function can be written as $g(x) = \alpha x + \beta$, what are $\alpha$ and $\beta$?

That’s a function equality problem. Two functions are equal if they give the same output for every input in the domain. So the linear expression $\alpha x + \beta$ must match the given pairs exactly. That means for each $x$ in $\{1,2,3,4\}$, we must have:

We don’t need all four — two pairs are enough to determine two unknowns. But we should check consistency with the rest.

Step-by-step solution

1. Confirm it’s a function.

Each $x$ in $\{1,2,3,4\}$ appears exactly once as a first element. No $x$ maps to two different $y$’s. So $g$ is indeed a function.

2. Set up the linear form.

We assume $g(x) = \alpha x + \beta$. Using the first two pairs:

• From $(1,1)$: $\alpha(1) + \beta = 1 \quad\Rightarrow\quad \alpha + \beta = 1$
• From $(2,3)$: $\alpha(2) + \beta = 3 \quad\Rightarrow\quad 2\alpha + \beta = 3$

3. Solve the system.

Subtract the first equation from the second:

Then $\alpha + \beta = 1$ gives $2 + \beta = 1$, so $\beta = -1$.

4. Verify with the remaining pairs.

Check $(3,5)$: $g(3) = 2(3) - 1 = 6 - 1 = 5$ ✓

Check $(4,7)$: $g(4) = 2(4) - 1 = 8 - 1 = 7$ ✓

All four points lie on the same straight line.

Method: Confirming a set is a function, then fitting a linear rule

Two tasks: verify the pairs form a function, then determine the coefficients of an assumed linear form.

Steps

Step 1: Function test

Check that each first coordinate appears once with a single image — no input repeats with different outputs.

Step 2: Set up equations from two data points

Assume $g(x)=\alpha x+\beta$ and substitute two known pairs to get two linear equations in $\alpha,\beta$.

Step 3: Solve the system

Subtract one equation from the other to eliminate $\beta$ and find $\alpha$; back-substitute for $\beta$.

Step 4: Verify with the remaining points

Plug the leftover pairs into $g(x)=\alpha x+\beta$; if they fit, the linear model is valid (constant slope confirms linearity).

Two points fix a line, but always confirm the others lie on it — otherwise the data is not linear.

Common Mistakes

Mistake 1: Assuming any table of pairs must be linear

Why it's wrong: a linear fit is only valid if the slope between consecutive points is constant. Correct approach: after solving for $\alpha,\beta$, check the remaining pairs actually satisfy $g(x)=\alpha x+\beta$.

Mistake 2: Sign slip solving for $\beta$

Why it's wrong: from $\alpha+\beta=1$ with $\alpha=2$, we get $\beta=-1$, not $+1$. Correct approach: substitute $\alpha$ back carefully.

Mistake 3: Skipping the "is it a function?" step

Why it's wrong: the question first asks whether $g$ is a function; the linear fit is meaningless otherwise. Correct approach: confirm each input has exactly one output before finding $\alpha,\beta$.