Certainly! Let's break down the function [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex] step by step:
1. Function Definition:
- The function [tex]\( f(x) \)[/tex] is defined as [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex].
2. Function Components:
- [tex]\( \frac{3}{4} \)[/tex] is the coefficient of [tex]\( x \)[/tex].
- [tex]\( 5 \)[/tex] is a constant term.
3. Linear Function:
- This function is linear because it can be expressed in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope ([tex]\( m \)[/tex]) is [tex]\( \frac{3}{4} \)[/tex], and the y-intercept ([tex]\( b \)[/tex]) is [tex]\( 5 \)[/tex].
4. Determining Values:
- To find the value of the function for specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \frac{3}{4} \cdot 0 + 5 = 5
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = \frac{3}{4} \cdot 4 + 5 = 3 + 5 = 8
\][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = \frac{3}{4} \cdot (-4) + 5 = -3 + 5 = 2
\][/tex]
5. Graphing the Function:
- The slope [tex]\( \frac{3}{4} \)[/tex] indicates that for every unit increase in [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] increases by [tex]\( \frac{3}{4} \)[/tex].
- The y-intercept [tex]\( 5 \)[/tex] indicates that the function crosses the y-axis at [tex]\( (0, 5) \)[/tex].
6. Example of Evaluating the Function:
- Let's find [tex]\( f(10) \)[/tex]:
[tex]\[
f(10) = \frac{3}{4} \cdot 10 + 5 = \frac{30}{4} + 5 = 7.5 + 5 = 12.5
\][/tex]
- Let's find [tex]\( f(-10) \)[/tex]:
[tex]\[
f(-10) = \frac{3}{4} \cdot (-10) + 5 = -\frac{30}{4} + 5 = -7.5 + 5 = -2.5
\][/tex]
By understanding each component of [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex] and performing evaluations for different values of [tex]\( x \)[/tex], we gain a thorough understanding of the behavior and characteristics of this linear function.
