If the range of the function f(x)=x2−3x+25−x, x=1,2, is (−∞,α)∪[β,∞), then α2+β2 is equal to:
Get the complete, step-by-step math solution for: "If the range of the function f(x) = (5 - x)/(x² - 3x + 2), x ≠ 1{,}2, is ( -∞, α) [β, ∞), then α² + β² is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Rewrite the function and set up for range calculation
To find the range of the function, we first set y=f(x). The denominator can be factored as x2−3x+2=(x−1)(x−2). So the function is y=(x−1)(x−2)5−x. We need to express x in terms of y to find the possible values of y.
Step 2: Rearrange into a quadratic equation in x
Multiply both sides by the denominator (x-1)(x-2) to get y(x2−3x+2)=5−x. Rearranging the terms to form a quadratic equation in x, we get yx2−3yx+2y=5−x, which simplifies to yx2+(−3y+1)x+(2y−5)=0.
Step 3: Apply the discriminant condition for real roots
For x to be a real number, the discriminant (D) of this quadratic equation must be greater than or equal to zero. The discriminant is given by D=b2−4ac, where a=y, b=(−3y+1), and c=(2y−5).
Step 4: Solve the inequality for y
Expanding the discriminant, we get (9y2−6y+1)−(8y2−20y)≥0. This simplifies to 9y2−6y+1−8y2+20y≥0, which further simplifies to y2+14y+1≥0.
Step 5: Find the roots of the quadratic and determine the range
To solve y2+14y+1≥0, we find the roots of y2+14y+1=0 using the quadratic formula. The roots are y=−7+43 and y=−7−43. Since the parabola y2+14y+1 opens upwards, the inequality y2+14y+1≥0 holds when y≤−7−43 or y≥−7+43. Thus, α=−7−43 and β=−7+43.
Step 6: Calculate α2+β2
Now we need to calculate α2+β2. We have α=−7−43 and β=−7+43. Using the identity (a−b)2+(a+b)2=2(a2+b2), with a=−7 and b=43, we can simplify the calculation.
Step 7: Simplify the expression
Substituting the values into the identity, we get 2((−7)2+(43)2)=2(49+16×3)=2(49+48)=2(97)=194.