Questions: Which of the following functions is not a function? (f(x)=x) (g(x)=2x+1) (h(x)=-3x^3+2) (j(x)=5x+3) if (x>5) and (j(x)=5) if (x<0) (k(x)=5x+3) if (x>1) and (k(x)=2x) if (x>-1)

Transcript text: Which of the following functions is not a function? $f(x)=$ $x$ \[ \begin{array}{l} g(x)=2 x \\ +1 \end{array} \] \[ \begin{array}{l} h(x)=-3 x^{3} \\ +2 \end{array} \] \[ \begin{array}{l} j(x)=5 x+3 \text { if } x>5 \text { and } j(x) \\ =5 \text { if } x<0 \end{array} \] $k(x)=5 x+3$ if $x>1$ and $k(x)=$ $2 x$ if $x>-1$

Solution

Solution Steps

To determine which of the given options is not a function, we need to check if each option assigns exactly one output for every input in its domain. A function can have different expressions for different parts of its domain, but each input must map to a single output.

$ f(x) = x $: This is a linear function and assigns exactly one output for every input.
$ g(x) = 2x + 1 $: This is also a linear function and assigns exactly one output for every input.
$ h(x) = -3x^3 + 2 $: This is a cubic function and assigns exactly one output for every input.
$ j(x) $: This piecewise function assigns different outputs based on the input value, but it still assigns exactly one output for each input.
$ k(x) $: This piecewise function has overlapping conditions for $ x > 1 $ and $ x > -1 $, which can lead to multiple outputs for the same input, making it not a function.

Step 1: Understanding the Problem

We are given several expressions and need to determine which one is not a function. A function is defined as a relation where each input (usually $x$) has exactly one output. We will analyze each given expression to see if it meets this criterion.

Step 2: Analyze Each Expression

$f(x) = x$
- This is a linear function where each input $x$ maps to exactly one output $x$. Therefore, $f(x) = x$ is a function.
$g(x) = 2x + 1$
- This is also a linear function. For each input $x$, there is a unique output $2x + 1$. Therefore, $g(x) = 2x + 1$ is a function.
$h(x) = -3x^3 + 2$
- This is a cubic function. For each input $x$, there is a unique output $-3x^3 + 2$. Therefore, $h(x) = -3x^3 + 2$ is a function.
$j(x) = 5x + 3$ if $x > 5$ and $j(x) = 5$ if $x < 0$
- This is a piecewise function. For $x > 5$, the function is defined as $5x + 3$, and for $x < 0$, it is defined as 5. There is no overlap in the domain, so each input $x$ has exactly one output. Therefore, $j(x)$ is a function.
$k(x) = 5x + 3$ if $x > 1$ and $k(x) = 2x$ if $x > -1$
- This is also a piecewise function. However, there is an overlap in the domain for $x > 1$ and $x > -1$. For example, if $x = 2$, it could be mapped to both $5x + 3$ and $2x$, which means it does not have a unique output for each input. Therefore, $k(x)$ is not a function.