# Which of the following is a linear function?

This question aims to find the linear functions that have one or more variables and represents a straight-line graph. A linear function represents a polynomial function whose degree is either $0$ or $1$. The variable $x$ is the independent variable that increases along the x-axis, whereas the variable $y$ is the dependent variable that increases along the y-axis. The equation of linear function is also called a line equation or linear equation. It has the following equation:

$f(x) = ax + b$

Where $a$ is the exponent of $x$ and $x$ is an independent variable and $b$ is the constant. The value of the function $f(x)$ is dependent on the $ax$ + $b$ equation.

To make a linear graph,

• We need to plot the two points on the XY-axis
• Join two points with a straight line
• This straight line will indicate the linear equation.

Figure 1

In the above graph, the function is $f(x)$= $3x$ which means the slope is $a$ = $3$ and the $b$ intercept is $0$.

A linear equation has an expression that is used to plot the slope of the graph. This expression is called the slope formula, where $m$ represents a slope, $c$ represents an intercept, and $(x,y)$ represents the coordinates. The slope formula is written as:

$y = mx + c$

## Numerical Solution

The given linear functions are:

$a) f(x) = 3$

$f(x) = y$

Putting values in the formula:

$y = 0x + 3$

In this expression, the slope $m$ is $0$ and the $c$ intercept is $3$. Hence, it is a linear function.

$b) g(x) = 5 – 2x$

$g(x) = y$

Rearranging the equation and putting the values in the slope formula:

$y = -2x + 5$

In this expression, the slope $m$ is $-2$, and the $c$ intercept is $5$, which means it is a linear function.

$c) h(x) = \frac{2}{x} + 3$

The above expression does not satisfy the slope formula as $x$ is present in the denominator. Hence, it is not a linear function.

$d) t(x) = 5(x – 2)$

By using the distributive property, we can write the expression as:

$t(x) = 5x – 10$

$t(x) = y$

$y = 5x – 10$

In this expression, the slope $m$ is $5$ and $c$ intercept is $-10$. Hence, it is a linear function.

## Example

There are two functions $f(2)$ = $3$ and $f(3)$ = $4$. In these two functions, we can evaluate their ordered pairs as:

$(2 , 3) (3 , 4)$

$(x_1 , y_1) (x_2 , y_2)$

By slope formula:

$\frac{y_2 – y_1}{x_2 – x_1}$

$= \frac{4 – 3}{3 – 2}$

$= \frac{1}{1}$

The value of slope $m$ is $1$.

Image/Mathematical drawings are created in Geogebra.

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