Question

A verbal description of a linear function $f$is given. Express the function $f$in the form $f\left(x\right)=ax+b$.

The graph of the linear function $h$has slope $\frac{1}{2}$and $y$-intercept $3$.

Short answer

The required function is $h\left(x\right)=\frac{1}{2}x+3$.

Step-by-step solution

Step 1. Definition of linear function.

In a linear function, each input value has a unique output value. In other words, for each point in domain, there is a unique value in the range of function. It can be written as:

$f\left(x\right)=ax+b$ … (i)

Where$a$ is the slope and$b$ is the $y$-intercept, and both are real numbers.

Step 2. Given information.

The graph of the linear function$h$ has slope$\frac{1}{2}$ and $y$-intercept $3$.

Step 3. Write the function.

The slope of the graph of a linear function is $\frac{1}{2}$, so $a=\frac{1}{2}$. The intercept is $3$, so $b=3$.

Substitute$a=\frac{1}{2}$ and$b=3$ in (i) to find the function $h$.

$\begin{array}{l}h\left(x\right)=\left(\frac{1}{2}\right)x+\left(3\right)\\ h\left(x\right)=\frac{1}{2}x+3\end{array}$

Thus, the required function is $h\left(x\right)=\frac{1}{2}x+3$.