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Math  /  Algebra

QuestionQuanto Griffin HW Units 9.2 Question 9, 9.2.29 HW Score: 27\%, 5.4 of 20 points Points: 0 of 1 Save list
Use the horizontal line test to determine whether the function is one-to-one. f(x)=8x2−5f(x)=\frac{8}{x^{2}-5}
Is the function one-to-one? Yes No [−10,10,−10,10]XXcl=1Yscl=1\begin{array}{l} {[-10,10,-10,10] X \mathrm{Xcl}=1} \\ \mathrm{Yscl}=1 \end{array}

Studdy Solution

STEP 1

1. The function given is f(x)=8x2−5 f(x) = \frac{8}{x^2 - 5} .
2. A function is one-to-one if and only if no horizontal line intersects its graph more than once.
3. The horizontal line test is used to determine if a function is one-to-one.
4. The domain of the function is determined by the values of x x for which the denominator is not zero.

STEP 2

1. Identify the domain of the function.
2. Analyze the behavior of the function.
3. Apply the horizontal line test.
4. Determine if the function is one-to-one.

STEP 3

Identify the domain of the function f(x)=8x2−5 f(x) = \frac{8}{x^2 - 5} .
The function is undefined where the denominator is zero:
x2−5=0 x^2 - 5 = 0 x2=5 x^2 = 5 x=±5 x = \pm \sqrt{5}
Thus, the domain of f(x) f(x) is all real numbers except x=5 x = \sqrt{5} and x=−5 x = -\sqrt{5} .

STEP 4

Analyze the behavior of the function:
The function f(x)=8x2−5 f(x) = \frac{8}{x^2 - 5} is a rational function. The denominator x2−5 x^2 - 5 is always positive or negative except at x=±5 x = \pm \sqrt{5} .
As x x approaches ±5 \pm \sqrt{5} , the function tends to infinity or negative infinity, indicating vertical asymptotes at these points.

STEP 5

Apply the horizontal line test:
For a function to be one-to-one, any horizontal line should intersect the graph of the function at most once.
Consider the behavior of f(x) f(x) as x x approaches positive or negative infinity. The function approaches zero but never actually reaches it.
Since the function is symmetric about the y-axis and has vertical asymptotes, horizontal lines will intersect the graph more than once.

STEP 6

Determine if the function is one-to-one:
Based on the horizontal line test, the function f(x)=8x2−5 f(x) = \frac{8}{x^2 - 5} is not one-to-one because horizontal lines can intersect the graph more than once.
The answer is: No, the function is not one-to-one.

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