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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 $0$.

Expert Answer

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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