Which of the following functions are one-to-one? Select all that apply.

Question

(2 points)
Responses

lower f left parenthesis x right parenthesis equals Start Fraction x minus 1 over 3 x plus 3 End Fraction
Image with alt text: lower f left parenthesis x right parenthesis equals Start Fraction x minus 1 over 3 x plus 3 End Fraction

lower f left parenthesis x right parenthesis equals Root 5 x plus 9 End Root
Image with alt text: lower f left parenthesis x right parenthesis equals Root 5 x plus 9 End Root

lower f left parenthesis x right parenthesis equals Start Fraction 7 over 4 x squared End Fraction
Image with alt text: lower f left parenthesis x right parenthesis equals Start Fraction 7 over 4 x squared End Fraction

lower f left parenthesis x right parenthesis equals one half x cubed
Image with alt text: lower f left parenthesis x right parenthesis equals one half x cubed

lower f left parenthesis x right parenthesis equals 3 x superscript 4 baseline plus 7 x cubed

Answer 1

To determine which of the given functions are one-to-one, we need to check whether each function satisfies the definition of a one-to-one function: a function \( f(x) \) is one-to-one if, whenever \( f(a) = f(b) \), it implies that \( a = b \).

We will analyze each function one by one:

1. \( f(x) = \frac{x - 1}{3x + 3} \): This function is a rational function. To check if it is one-to-one, we could set \( f(a) = f(b) \) and see if it leads to \( a = b \). The denominator can change the nature of the function, especially around points where it might be undefined. A detailed approach via derivative could also be used. In general, rational functions can often be one-to-one unless they present a repeating pattern, but without further complex analysis, it's hard to conclude. Generally, it does not guarantee one-to-oneness.

2. \( f(x) = \sqrt{5x + 9} \): This is a square root function. Since the square root function is monotonically increasing (it always increases as \( x \) increases), this function is one-to-one.

3. \( f(x) = \frac{7}{4} x^2 \): This is a quadratic function. Quadratic functions are not one-to-one unless restricted to a domain where they are either entirely increasing or decreasing (i.e., the vertex point). Therefore, this function is not one-to-one as it can have the same output for different inputs.

4. \( f(x) = \frac{1}{2} x^3 \): This function is a cubic function. Cubic functions are one-to-one across their entire domain because they are continuous and have only one real root (the inflection point). Therefore, this function is one-to-one.

5. \( f(x) = 3x^4 + 7x^3 \): This is a polynomial function of degree 4. Polynomial functions of degree greater than 1 can have multiple outputs for the same input, usually if they can "turn back" on themselves. This function can be checked for being one-to-one, but generally, quartics can exhibit behavior making them not one-to-one. The derivative also can show potential critical points (which could indicate local minima or maxima), potentially leading to different inputs giving the same output. As such, this function is not guaranteed to be one-to-one.

Summary:

From this analysis, the one-to-one functions are:

• \( f(x) = \sqrt{5x + 9} \)
• \( f(x) = \frac{1}{2}x^3 \)

Therefore, the selections are:

• \( f(x) = \sqrt{5x + 9} \)
• \( f(x) = \frac{1}{2}x^3 \)