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Zorluk: OrtaFunction Evaluation, Composition, and Properties

Two functions are defined as f(x)=(x−3)2−16f(x) = (x - 3)^2 - 16 and g(x)=∣x−1∣−8g(x) = |x - 1| - 8. If g(f(x))=0g(f(x)) = 0, what is the sum of all positive values of xx?

  1. A
    44
  2. B
    88
  3. C
    1010
  4. 1414Cevap
  5. E
    2424

Cevap

The correct sum of all positive values of xx is 14.
To find the sum of all positive values of xx that satisfy g(f(x))=0g(f(x)) = 0, first substitute f(x)f(x) into g(x)g(x) to get the equation ∣f(x)−1∣−8=0|f(x) - 1| - 8 = 0. This simplifies to ∣f(x)−1∣=8|f(x) - 1| = 8. Solving this absolute value equation yields two cases: f(x)−1=8f(x) - 1 = 8 or f(x)−1=−8f(x) - 1 = -8.

For the first case, f(x)=9f(x) = 9. Substituting f(x)=(x−3)2−16f(x) = (x - 3)^2 - 16 gives (x−3)2−16=9⇒(x−3)2=25(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25. Taking the square root of both sides gives x−3=5x - 3 = 5 or x−3=−5x - 3 = -5, which results in x=8x = 8 or x=−2x = -2. The only positive solution from this case is 88.

For the second case, f(x)=−7f(x) = -7. Substituting f(x)=(x−3)2−16f(x) = (x - 3)^2 - 16 gives (x−3)2−16=−7⇒(x−3)2=9(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9. Taking the square root of both sides gives x−3=3x - 3 = 3 or x−3=−3x - 3 = -3, which results in x=6x = 6 or x=0x = 0. The only positive solution from this case is 66 (since 00 is not positive).

Adding the positive solutions gives 8+6=148 + 6 = 14.

Adım Adım Çözüm

1
Set up the composite equation g(f(x))=0g(f(x)) = 0.
∣f(x)−1∣−8=0⇒∣f(x)−1∣=8|f(x) - 1| - 8 = 0 \Rightarrow |f(x) - 1| = 8
Substitute the expression for f(x)f(x) into g(x)g(x) to establish the relationship.
2
Solve the absolute value equation by separating it into two distinct cases.
f(x)−1=8⇒f(x)=9f(x) - 1 = 8 \Rightarrow f(x) = 9 or f(x)−1=−8⇒f(x)=−7f(x) - 1 = -8 \Rightarrow f(x) = -7
An absolute value equation ∣u∣=c|u| = c splits into u=cu = c and u=−cu = -c when c≥0c \geq 0.
3
Substitute the definition of f(x)=(x−3)2−16f(x) = (x - 3)^2 - 16 into the first case and solve for xx.
(x−3)2−16=9⇒(x−3)2=25⇒x−3=±5⇒x=8(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25 \Rightarrow x - 3 = \pm 5 \Rightarrow x = 8 or x=−2x = -2
Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.
4
Substitute the definition of f(x)=(x−3)2−16f(x) = (x - 3)^2 - 16 into the second case and solve for xx.
(x−3)2−16=−7⇒(x−3)2=9⇒x−3=±3⇒x=6(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9 \Rightarrow x - 3 = \pm 3 \Rightarrow x = 6 or x=0x = 0
Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.
5
Identify the positive solutions and calculate their sum.
The positive solutions are 88 and 66. The sum is 8+6=148 + 6 = 14.
Exclude non-positive values (−2-2 is negative, and 00 is neither positive nor negative) and add the remaining values.

Anahtar Kavram

Function composition involves substituting one function into another, and evaluating the resulting composite equation requires solving multi-step equations including absolute value and quadratic relations.
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Konular

Function Evaluation, Composition, and Properties

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