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.