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

Question

Responses

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

g(x)=2x2
𝑔
(
𝑥
)
=
2
𝑥
2
g left parenthesis x right parenthesis equals 2 x squared

g(x)=−6x4+2x−1
𝑔
(
𝑥
)
=
−
6
𝑥
4
+
2
𝑥
−
1
g left parenthesis x right parenthesis equals negative 6 x superscript 4 baseline plus 2 x minus 1

g(x)=x2−4x+1

Answer 1

To determine which of the given functions is one-to-one, we can analyze them individually. A function is one-to-one if it assigns a unique output for every unique input, meaning \(g(a) = g(b)\) implies \(a = b\).

1. g(x) = 2x - 4

   • This is a linear function with a non-zero slope (2). Linear functions with a slope not equal to zero are one-to-one.

2. g(x) = 2x²

   • This is a quadratic function. Quadratic functions are generally not one-to-one because they can produce the same output for different inputs (for example, \(g(1) = g(-1) = 2\)).

3. g(x) = -6x⁴ + 2x - 1

   • This is a polynomial function of degree 4 (even degree) and has a higher degree term that is negative. Polynomial functions of even degree are typically not one-to-one since they can have turning points and exhibit symmetry.

4. g(x) = x² - 4x + 1

   • This is a quadratic function that can also be expressed as \(g(x) = (x - 2)² - 3\). As with the previous quadratic function, this one is not one-to-one because it can also yield the same output for multiple inputs.

From this analysis:

• g(x) = 2x - 4 is the only one-to-one function among the given options.