To determine if each of the provided functions is one-to-one, we will analyze them based on whether they pass the horizontal line test or if they have unique outputs for every input throughout their domain.
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\( f(x) = \frac{x-1}{3x+3} \): This function is a rational function. It can be shown to be one-to-one using calculus by finding the derivative and confirming that it is either always positive or always negative in its domain, confirming monotonicity. Calculating the derivative will offer insight, but since rational functions generally behave well in terms of one-to-one nature, this function is likely one-to-one. Therefore, this function is one-to-one.
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\( f(x) = \sqrt{5x + 9} \): The square root function is one-to-one because it only takes non-negative values for the input it accepts, meaning for any two different inputs \( x_1 \) and \( x_2 \), if \( x_1 \neq x_2 \) then \( \sqrt{5x_1 + 9} \neq \sqrt{5x_2 + 9} \). Thus, this function is one-to-one.
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\( f(x) = \frac{7}{4} x^2 \): This function is a quadratic function with a parabolic shape that opens upwards. It fails the horizontal line test, meaning there are inputs \( x_1 \) and \( x_2 \) (such as \( x_1 = 1 \) and \( x_2 = -1 \)) that give the same output. Therefore, this function is not one-to-one.
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\( f(x) = \frac{1}{2} x^3 \): This function is a cubic function and is continuous and increasing over its entire domain (i.e., it passes the horizontal line test). Since it is strictly increasing, this function is one-to-one.
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\( f(x) = 3x^4 + 7x^3 \): This function is a polynomial function of degree 4. Its derivative will reveal the critical points and whether it is strictly increasing or decreasing. Given that the higher degree term \( 3x^4 \) can cause this function to not be one-to-one, we would need to analyze its derivative. In fact, because it is a quartic equation, it can have local minima and maxima leading to additional intersections. Therefore, this function is generally known to not be one-to-one. This function is not one-to-one.
In summary, the functions which are one-to-one among the given options are:
- \( f(x) = \frac{x-1}{3x+3} \)
- \( f(x) = \sqrt{5x + 9} \)
- \( f(x) = \frac{1}{2} x^3 \)
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