Which of the following functions are one-to-one? Select all that apply. (2 points) Responses Image with alt text: 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 Root 5 x plus 9 End Root Image with alt text: 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 one half x cubed Image with alt text: lower f left parenthesis x right parenthesis equals 3 x superscript 4 baseline plus 7 x cubed
The functions that are one-to-one are:
1. $f(x) = \frac{x - 1}{3x + 3}$
2. $f(x) = \sqrt{5x + 9}$
3. $f(x) = \frac{7}{4}x^2$
The functions that are not one-to-one are:
1. $f(x) = \frac{1}{2}x^3$
2. $f(x) = 3x^4 + 7x^3$
To determine which functions are one-to-one, we need to check if each function has a unique output for every input.
1. lower f(x) = (x - 1)/(3x + 3)
To check if this function is one-to-one, we need to see if different inputs yield different outputs. However, since there is no image or equation for the function provided, we cannot determine if it is one-to-one.
2. lower f(x) = √(5x + 9)
This function represents a square root of an expression. Since a square root function has the property that different inputs yield different outputs, this function is one-to-one.
3. lower f(x) = (7/4)x^2
In this function, the coefficient in front of x^2 is positive, which means the function is always increasing or always decreasing. Therefore, different inputs result in different outputs, making it a one-to-one function.
4. lower f(x) = (1/2)x^3
Similar to the previous function, the coefficient in front of x^3 is positive, which indicates that this function is always increasing or always decreasing. Thus, it is one-to-one.
5. lower f(x) = 3x^4 + 7x^3
Just like the previous two functions, this function also has a positive coefficient in front of the highest power term (x^4). Therefore, it is also one-to-one.
Based on our analysis, the functions that are one-to-one are:
2. lower f(x) = √(5x + 9)
3. lower f(x) = (7/4)x^2
4. lower f(x) = (1/2)x^3
5. lower f(x) = 3x^4 + 7x^3
To determine which of the given functions are one-to-one, we need to evaluate their inverse functions. If the inverse function exists and is also a function, then the original function is one-to-one.
Let's evaluate the inverse functions for each of the given functions:
1. lower f(x) = (x - 1) / (3x + 3)
To find the inverse function, let's swap x and f(x) and solve for f(x):
x = (f(x) - 1) / (3f(x) + 3)
Multiply both sides by (3f(x) + 3):
x(3f(x) + 3) = f(x) - 1
Expand and rearrange the equation:
3xf(x) + 3x = f(x) - 1
Subtract f(x) and 3x from both sides:
3xf(x) - f(x) = -3x - 1
Factor out f(x) on the left side:
f(x)(3x - 1) = -3x - 1
Divide both sides by (3x - 1):
f(x) = (-3x - 1) / (3x - 1)
The inverse function exists and is a function, so the original function is one-to-one.
2. lower f(x) = √(5x + 9)
To find the inverse function, let's swap x and f(x) and solve for f(x):
x = √(5f(x) + 9)
Square both sides to eliminate the square root:
x^2 = 5f(x) + 9
Subtract 9 from both sides:
x^2 - 9 = 5f(x)
Divide both sides by 5:
f(x) = (x^2 - 9) / 5
The inverse function exists and is a function, so the original function is one-to-one.
3. lower f(x) = (7/4)x^2
To find the inverse function, let's swap x and f(x) and solve for f(x):
x = (7/4)f(x)^2
Multiply both sides by 4/7 to isolate f(x)^2:
(4/7)x = f(x)^2
Take the square root of both sides:
√((4/7)x) = f(x)
The inverse function exists and is a function, so the original function is one-to-one.
4. lower f(x) = 1/2x^3
To find the inverse function, let's swap x and f(x) and solve for f(x):
x = 1/2f(x)^3
Multiply both sides by 2 to isolate f(x)^3:
2x = f(x)^3
Take the cube root of both sides:
∛(2x) = f(x)
The inverse function exists and is a function, so the original function is one-to-one.
5. lower f(x) = 3x^4 + 7x^3
To find the inverse function, let's swap x and f(x) and solve for f(x):
x = 3f(x)^4 + 7f(x)^3
This equation cannot be easily rearranged to solve for f(x) explicitly. Therefore, the inverse function may not exist, or it may not be a function. Therefore, the original function may not be one-to-one.
In summary:
The first four functions are one-to-one. The fifth function may not be one-to-one as its inverse function cannot be explicitly determined as a function.