Find the constants a and b such that the function is continuous on the entire line

f(x){7, x is less than or equal to -3
ax+b, -3 is less than x is greater than 4
-7, x is less than or equal to 4

you need a smooth transition from (-3,7) to (4,-7)

y-7 = -14/7 (x+3)
or
y = -2x + 1

so, a=-2, b=1

To find the constants a and b such that the function is continuous on the entire line, we need to find the values of a and b that make the function "smooth" at the points of transition.

First, let's consider the left side of the transition when x ≤ -3. The function is given as f(x) = 7 for x ≤ -3. This means that for any x value less than or equal to -3, the function should have a constant value of 7. So, the function is already continuous and has no dependency on a or b for x ≤ -3.

Next, let's consider the right side of the transition when -3 < x < 4. The function is given as f(x) = ax + b for -3 < x < 4. Now, we need to find the values of a and b that ensure the function is continuous at x = -3 and x = 4.

To make the function continuous at x = -3, the left-side limit and the right-side limit must be equal. The left-side limit is 7 (from the previous discontinuous function), and the right-side limit is obtained by plugging in x = -3 into the function f(x) = ax + b. Thus, we have:

7 = a(-3) + b

To make the function continuous at x = 4, the left-side limit and the right-side limit must be equal. The function is given as f(x) = -7 for x ≤ 4. Thus, we have:

-7 = a(4) + b

Now, we have two equations:

7 = -3a + b -- Equation 1
-7 = 4a + b -- Equation 2

Solving these equations simultaneously will give us the values of a and b. There are several methods to solve these equations, such as substitution or elimination. I will use the method of substitution to solve them.

From Equation 2, let's solve for b in terms of a:

b = -7 - 4a

Now, substitute this value of b into Equation 1:

7 = -3a + (-7 - 4a)

Simplify the equation:

7 = -7a - 7

Add 7 to both sides:

14 = -7a

Divide both sides by -7:

a = -2

Now, substitute the value of a back into Equation 1 to find b:

7 = -3(-2) + b

Simplify the equation:

7 = 6 + b

Subtract 6 from both sides:

1 = b

Therefore, the constants a and b that make the function continuous on the entire line are a = -2 and b = 1.

Thus, the final function is:

f(x) = 7, x ≤ -3
f(x) = -2x + 1, -3 < x < 4
f(x) = -7, x ≥ 4