Examkin

Question

Difficulty: MediumFunction Evaluation, Composition, and Properties

Let the functions ff and gg be defined for all real numbers by f(x)=2x−3f(x) = 2x - 3 and g(x)=x2−5g(x) = x^2 - 5. If g(f(x))=11g(f(x)) = 11 and x<0x < 0, what is the value of xx?

  1. A
    72\frac{7}{2}
  2. B
    −12-\sqrt{12}
  3. C
    −72-\frac{\sqrt{7}}{2}
  4. −12-\frac{1}{2}Answer
  5. E
    −72-\frac{7}{2}

Answer

−12-\frac{1}{2}
The correct answer is −12-\frac{1}{2}. First, the composite function g(f(x))g(f(x)) is found by substituting f(x)=2x−3f(x) = 2x - 3 into g(x)=x2−5g(x) = x^2 - 5, giving g(f(x))=(2x−3)2−5g(f(x)) = (2x - 3)^2 - 5. Setting this expression equal to 1111 yields the equation (2x−3)2−5=11(2x - 3)^2 - 5 = 11. Adding 55 to both sides results in (2x−3)2=16(2x - 3)^2 = 16. Taking the square root of both sides gives 2x−3=±42x - 3 = \pm 4. Since the problem specifies that x<0x < 0, the expression 2x−32x - 3 must be negative because 2x<02x < 0 and subtracting 33 makes the result less than −3-3. Thus, we set 2x−3=−42x - 3 = -4. Adding 33 to both sides yields 2x=−12x = -1, and dividing by 22 gives x=−12x = -\frac{1}{2}.

Step-by-Step Solution

1
Substitute the expression for f(x)f(x) into the function g(x)g(x) to obtain the composite function g(f(x))g(f(x)).
g(f(x))=g(2x−3)=(2x−3)2−5g(f(x)) = g(2x - 3) = (2x - 3)^2 - 5
To find g(f(x))g(f(x)), we substitute the entire function f(x)f(x) in place of the input variable in g(x)g(x).
2
Set the composite function equal to 1111 and isolate the squared term.
(2x−3)2−5=11⇒(2x−3)2=16(2x - 3)^2 - 5 = 11 \Rightarrow (2x - 3)^2 = 16
We are given that g(f(x))=11g(f(x)) = 11, and adding 55 to both sides isolates the squared binomial.
3
Solve for 2x−32x - 3 by taking the square root of both sides, applying the constraint x<0x < 0.
2x−3=−42x - 3 = -4
Since x<0x < 0, it follows that 2x<02x < 0, which means 2x−3<−32x - 3 < -3. Because 2x−32x-3 must be negative, we take the negative square root of 1616.
4
Solve the linear equation 2x−3=−42x - 3 = -4 for xx.
2x=−1⇒x=−122x = -1 \Rightarrow x = -\frac{1}{2}
Adding 33 to both sides gives 2x=−12x = -1, and dividing by 22 yields the final value of xx.

Key Concept

Function Composition and Solving Quadratic/Linear Equations
Estimated Time:1m 30s
Rate this question

Topics

Function Evaluation, Composition, and Properties

More questions from this topic