Question

Identify which of the following functions is one-to-one.(1 point)
Responses

g(x)=2x−4
g left parenthesis x right parenthesis equals 2 x minus 4

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)=x2−4x+1
g left parenthesis x right parenthesis equals x squared minus 4 x plus 1

g(x)=2x2

Answer 1

To determine which function is one-to-one, we need to check if each function passes the Horizontal Line Test, meaning that each output value corresponds to exactly one input value.

1. **\( g(x) = 2x - 4 \)**
- This is a linear function with a slope of 2. It is one-to-one because for every \( x \), there is a unique \( g(x) \).

2. **\( g(x) = -6x^4 + 2x - 1 \)**
- This is a polynomial function. The term \( -6x^4 \) dominates as \( x \) becomes large, making it not one-to-one. It will have multiple values of \( g(x) \) for some values of \( x \).

3. **\( g(x) = x^2 - 4x + 1 \)**
- This is a quadratic function. Quadratic functions are not one-to-one because they can yield the same output for different \( x \) values (e.g., \( g(0) = 1 \) and \( g(4) = 1 \)).

4. **\( g(x) = 2x^2 \)**
- This is another quadratic function. Similar to the previous one, it cannot be one-to-one because it also can have multiple inputs that yield the same output (e.g., \( g(1) = 2 \) and \( g(-1) = 2 \)).

From the analysis above, the only one-to-one function is:

**\( g(x) = 2x - 4 \)**.