Questions: Which of the following functions is not a one-to-one function? f(x)=x g(x)=2x+1 h(x)=-2x^2+2 j(x)=5x^3+4 k(x)=3x^5-4

Transcript text: Which of the following functions is not a one-to-one function? $f(x)=x$ $g(x)=2x+1$ $h(x)=-2x^{2}+2$ $j(x)=5x^{3}+4$ $k(x)=3x^{5}-4$

Solution

Solution Steps

To determine which of the given functions is not a one-to-one function, we need to check if each function passes the horizontal line test. A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once.

For polynomial functions:

Linear functions (degree 1) are always one-to-one.
Quadratic functions (degree 2) are not one-to-one because they are parabolas and can intersect a horizontal line at two points.
Higher-degree polynomials need to be checked individually, but odd-degree polynomials (like cubic and quintic) are typically one-to-one.

Given the functions:

$ f(x) = x $ is linear and one-to-one.
$ g(x) = 2x + 1 $ is linear and one-to-one.
$ h(x) = -2x^2 + 2 $ is quadratic and not one-to-one.
$ j(x) = 5x^3 + 4 $ is cubic and one-to-one.
$ k(x) = 3x^5 - 4 $ is quintic and one-to-one.

Thus, the function $ h(x) = -2x^2 + 2 $ is not a one-to-one function.

Step 1: Identify the Functions

We are given the following functions:

$ f(x) = x $
$ g(x) = 2x + 1 $
$ h(x) = -2x^2 + 2 $
$ j(x) = 5x^3 + 4 $
$ k(x) = 3x^5 - 4 $

Step 2: Determine One-to-One Functions

A function is one-to-one if it passes the horizontal line test, meaning that for any horizontal line $ y = c $, the equation $ f(x) = c $ has at most one solution.

For $ f(x) = x $ (linear), it is one-to-one.
For $ g(x) = 2x + 1 $ (linear), it is one-to-one.
For $ h(x) = -2x^2 + 2 $ (quadratic), it is not one-to-one because it opens downwards and can intersect a horizontal line at two points.
For $ j(x) = 5x^3 + 4 $ (cubic), it is one-to-one.
For $ k(x) = 3x^5 - 4 $ (quintic), it is one-to-one.

Step 3: Conclusion on One-to-One Status

From the analysis, we find that: