Easy Peasy Functions!

Since it is linear function, $f(x)$ satisfies $f(cx) = cf(x)$ . Then,

$\begin{aligned} 12 &= f(6) - f(2)\\ &= f(3(2)) - f(2)\\ &= 3f(2) - f(2)\\ &= 2f(2)\\ f(2) &= 6 \end{aligned}$

Therefore,

$\begin{aligned} f(12) - f(2)&= 6f(2) - f(2) \\ &= 5f(2) \\ &= 5(6) \\ &= \boxed{30} \\ \end{aligned}$

Well, linear functions are real functions which map x to ax+C for some real a and C

Assume the real function in this problem is f(x) = ax+C

We're given that f(6)-f(2)=12

This implies that f(6)-f(2)= 6a + C - (2a+C) = 4a = 12

Solving, we get a = 3

Now, look into the other part

f(12)-f(2)=12a+C -(2a+C) = 10a

Substitute, a with 3 and get the answer as 30

Since $f(x)$ is a linear function, then we can compute for its slope $\frac{f(x_1) - f(x_2)}{x_1 - x_2} = m$ .

Recomputing, we can have $f(x_1) - f(x_2) = m (x_1 - x_2)$ .

With this, we now have the following:

$f(6) -f(2) = 12 = m (6 - 2)$ which produces $m=3$ .

Thus, $f(12) -f(2) = m (12 - 2) = 3 (10) = 30$

• Definition of a linear function: f is called linear if it satisfies (i) f(cx) = cf(x) and (ii) f(x + y) = f(x) + f(y) for any vectors x, y and any scalar constant c.

So,

12 = f(6) - f(2) = f(6) + f(-2) = f(6 - 2) = f(4).

Now,

f(12) - f(2) = f(3 * 4) - f((1/2) * 4) = 3f(4) - (1/2)f(4) = 3 * 12 - (1/2) * 12 = 30.

Let 6x-2x=12 OR 4x=12...thus x=3.
Now,
to find 12x-2x = 12(3)-2(3) = 36-6 = 30

Let the function $f(x)$ be represented as $Ax+B=0$ , as it is the general form of linear algebraic functions.
So,
$f(6)=6A+B$
$f(2)=2A+B$
$f(6)-f(2)=(6A+B)-(2A+B)$
$12=6A+B-2A-B$
$12=4A$
$A=3$
So, $f(12)-f(2)=(12A+B)-(2A+B)$
$\Rightarrow f(12)-f(2)= 10A$
$\Rightarrow f(12)-f(2)= 30$

Linear function means st. line. Given change in y = 12 change in .x = 6 - 2 = 4.
Slope = 12/4 = 3. Change in x now is 12 - 2 = 10...
.Hence change in y = 3 * 10 = 30.

$f(6)-f(2)=12$ ... or,difference of every f(1)= $\frac{12}{4}=3$

so, $f(12)-f(2)=f(3 \times 12)-f(3 \times 2)=36-6=30$

f(x) is lineal so, when it goes from (2) to (6) takes 4 horizontal steps, and when goes from (2) to (12) takes 10. Then in order to get the second amount of vertical steps we must multiply the first amount, 12, by 10/4 = 2.5, then 24 + 6 = 30.

Write f(x) = ax + b since it is a linear equation,

f(6) - f(2) = 12 is equal to write

6a + b - 2a - b = 12 or

4a = 12

a = 3

so the function is f(x) = 3x + b .

Than f(12) - f(2) = 36 + b - 6 - b = 30

The definition of a slope is m = (y - y1) / (x - x1) or, ((f(x) - f(x1)) / (x - x1))

We can see what the x values are; for f(6) the x-value is 6, and for f(2) the x-value is 2.

Therefore: ((f(6) - f(2)) / (6 - 2)) = m. Given is that f(6) - f(2) = 12. The slope, m, is thus 12 / 4 = 3.

Since it is give that we are dealing with a linear function, we therefore know that ((f(12) - f(2)) / (12 - 2)) = 3, and so (f(12) - f(2)) = 10 * 3 = 30.

given ,f(x) is a linear function.then let it be ax+b. then,f(6)=6x+b,f(2)=2x+b. then f(6)-f(2)=(6x+b)-(2x+b) => 6x+b-2x-b=12 => 4x=12 => x=3 for f(12)-f(2)=12x+b-2x-b=12(3)-2(3)=36-6=30