Questions: Let f be a function with constant rate of change. (a) Then f is a function and f is of the form f(x)= x+ . (b) The graph of f is

Let f be a function with constant rate of change. (a) Then f is a function and f is of the form f(x)= x+ . (b) The graph of f is
Transcript text: Let $f$ be a function with constant rate of change. (a) Then $f$ is a $\square$ function and $f$ is of the form $f(x)=$ $\square$ $x+$ $\square$ . (b) The graph of $f$ is $\square$
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Solution

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Solution Steps

To solve this problem, we need to identify the type of function that has a constant rate of change. This type of function is a linear function. A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y-intercept. The graph of a linear function is a straight line.

Step 1: Identify the Type of Function

The function \( f \) has a constant rate of change, which indicates that it is a linear function. A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Determine the Form of the Function

Given the form \( f(x) = mx + b \), we can substitute the values \( m = 2 \) and \( b = 3 \) into the equation. Thus, the function becomes: \[ f(x) = 2x + 3 \]

Step 3: Calculate the Value of the Function

To find the value of the function at \( x = 5 \), substitute \( x = 5 \) into the equation: \[ f(5) = 2(5) + 3 = 10 + 3 = 13 \]

Final Answer

(a) The function \( f \) is a \(\boxed{\text{linear}}\) function and \( f(x) = \boxed{2x + 3} \).

(b) The graph of \( f \) is a \(\boxed{\text{straight line}}\).

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