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Math  /  Algebra

QuestionDetermine the real values of xx that cause the function to be (a) zero, (b) undefined, (c) positive, f(x)=xx+12f(x)=x \sqrt{x+12} A. x=−12x=-12 B. x<12x<12 C. x<−12x<-12 D. x=12x=12 (c) What value(s) of x cause the function to be positive? Choose the correct answer below. A. (∞,−12)∪(12,∞)(\infty,-12) \cup(12, \infty) B. (−12,∞)(-12, \infty) C. (12,∞)(12, \infty) D. (0,∞)(0, \infty)

Studdy Solution

STEP 1

1. The function f(x)=xx+12 f(x) = x \sqrt{x+12} involves a square root, which requires the radicand to be non-negative.
2. The function is zero when f(x)=0 f(x) = 0 .
3. The function is undefined when the radicand is negative.
4. The function is positive when f(x)>0 f(x) > 0 .

STEP 2

1. Determine when the function is zero.
2. Determine when the function is undefined.
3. Determine when the function is positive.

STEP 3

To find when the function is zero, set f(x)=0 f(x) = 0 :
xx+12=0 x \sqrt{x+12} = 0
This equation is zero if either x=0 x = 0 or x+12=0 \sqrt{x+12} = 0 .

STEP 4

Solve x+12=0 \sqrt{x+12} = 0 :
x+12=0 x + 12 = 0 x=−12 x = -12
The function is zero at x=0 x = 0 and x=−12 x = -12 .

STEP 5

To find when the function is undefined, determine when the radicand is negative:
x+12<0 x + 12 < 0 x<−12 x < -12
The function is undefined for x<−12 x < -12 .

STEP 6

To find when the function is positive, determine when both x>0 x > 0 and x+12≥0 x + 12 \geq 0 :
1. x>0 x > 0 ensures x x is positive.
2. x+12≥0 x + 12 \geq 0 simplifies to x≥−12 x \geq -12 .
The intersection of these conditions is x>0 x > 0 .
The correct answer for when the function is positive is:
D. (0,∞)(0, \infty)

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