Question
Identify which of the following functions is one-to-one.(1 point)
Responses
g(x)=2x2
g left parenthesis x right parenthesis equals 2 x squared
g(x)=x2−4x+1
g left parenthesis x right parenthesis equals x squared minus 4 x plus 1
g(x)=−6x4+2x−1
g left parenthesis x right parenthesis equals negative 6 x superscript 4 baseline plus 2 x minus 1
g(x)=2x−4
Responses
g(x)=2x2
g left parenthesis x right parenthesis equals 2 x squared
g(x)=x2−4x+1
g left parenthesis x right parenthesis equals x squared minus 4 x plus 1
g(x)=−6x4+2x−1
g left parenthesis x right parenthesis equals negative 6 x superscript 4 baseline plus 2 x minus 1
g(x)=2x−4
Answers
Answered by
GPT-4o mini
To determine which of the given functions is one-to-one, we need to check if each function passes the horizontal line test or if it can be shown algebraically that different inputs produce different outputs. A function \( g(x) \) is one-to-one if for any \( a \) and \( b \) in the domain, \( g(a) = g(b) \) implies that \( a = b \).
1. **\( g(x) = 2x^2 \)**:
- This is a quadratic function that opens upwards. Since it takes the same value for both \( x \) and \( -x \) (for example, \( g(1) = 2 \) and \( g(-1) = 2 \)), it is not one-to-one.
2. **\( g(x) = x^2 - 4x + 1 \)**:
- This is also a quadratic function. Completing the square gives \( g(x) = (x-2)^2 - 3 \). Since it is a parabola that opens upwards, it will also fail the horizontal line test and is not one-to-one.
3. **\( g(x) = -6x^4 + 2x - 1 \)**:
- This function is a quartic polynomial. The leading coefficient is negative, meaning that the function can change direction multiple times, which suggests it may not be one-to-one. However, for a definitive conclusion, we would analyze its derivative or examine its critical points. It's complex enough that it may have multiple outputs for some inputs, so we should assume it is not one-to-one.
4. **\( g(x) = 2x - 4 \)**:
- This is a linear function, and linear functions of the form \( ax + b \) where \( a \neq 0 \) are always one-to-one. For any \( a \) and \( b \), if \( g(a) = g(b) \), then \( 2a - 4 = 2b - 4 \) implies \( a = b \).
Based on this analysis, the only function that is one-to-one is:
**\( g(x) = 2x - 4 \)**.
1. **\( g(x) = 2x^2 \)**:
- This is a quadratic function that opens upwards. Since it takes the same value for both \( x \) and \( -x \) (for example, \( g(1) = 2 \) and \( g(-1) = 2 \)), it is not one-to-one.
2. **\( g(x) = x^2 - 4x + 1 \)**:
- This is also a quadratic function. Completing the square gives \( g(x) = (x-2)^2 - 3 \). Since it is a parabola that opens upwards, it will also fail the horizontal line test and is not one-to-one.
3. **\( g(x) = -6x^4 + 2x - 1 \)**:
- This function is a quartic polynomial. The leading coefficient is negative, meaning that the function can change direction multiple times, which suggests it may not be one-to-one. However, for a definitive conclusion, we would analyze its derivative or examine its critical points. It's complex enough that it may have multiple outputs for some inputs, so we should assume it is not one-to-one.
4. **\( g(x) = 2x - 4 \)**:
- This is a linear function, and linear functions of the form \( ax + b \) where \( a \neq 0 \) are always one-to-one. For any \( a \) and \( b \), if \( g(a) = g(b) \), then \( 2a - 4 = 2b - 4 \) implies \( a = b \).
Based on this analysis, the only function that is one-to-one is:
**\( g(x) = 2x - 4 \)**.