Question

1. slide red circles in front of correct answer. determine all of the following functions which have a range {y | y ≥ -5}. a. y=(x - 5)^2 + 3 c. y = x^3 - x^2 + 4 e. y = √(x - 5) + 3 b. y = √(x + 6) - 5 d. y + 5 = (x + 2)^2 f. y = x^2 - 5

Explanation:

Step1: Analyze function A

For \(y=(x - 5)^{2}+3\), since \((x - 5)^{2}\geq0\), then \(y=(x - 5)^{2}+3\geq3\). The range is \(\{y|y\geq3\}\), so it does not meet the requirement.

Step2: Analyze function B

For \(y=\sqrt{x + 6}-5\), because \(\sqrt{x+6}\geq0\), then \(y=\sqrt{x + 6}-5\geq- 5\). The range is \(\{y|y\geq - 5\}\), so it meets the requirement.

Step3: Analyze function C

For \(y=x^{4}-x^{2}+4\), let \(t=x^{2}(t\geq0)\), then \(y=t^{2}-t + 4=(t-\frac{1}{2})^{2}+\frac{15}{4}\). When \(t=\frac{1}{2}\), \(y=\frac{15}{4}\). The range is \(\{y|y\geq\frac{15}{4}\}\), so it does not meet the requirement.

Step4: Analyze function D

For \(y + 5=(x + 2)^{2}\), then \(y=(x + 2)^{2}-5\). Since \((x + 2)^{2}\geq0\), then \(y=(x + 2)^{2}-5\geq-5\). The range is \(\{y|y\geq - 5\}\), so it meets the requirement.

Step5: Analyze function E

For \(y=\sqrt{x - 5}+3\), because \(\sqrt{x-5}\geq0\), then \(y=\sqrt{x - 5}+3\geq3\). The range is \(\{y|y\geq3\}\), so it does not meet the requirement.

Step6: Analyze function F

For \(y=x^{2}-5\), since \(x^{2}\geq0\), then \(y=x^{2}-5\geq-5\). The range is \(\{y|y\geq - 5\}\), so it meets the requirement.

Answer:

B. \(y=\sqrt{x + 6}-5\), D. \(y + 5=(x + 2)^{2}\), F. \(y=x^{2}-5\)