To find the domain and range of the given function [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex], we need to analyze its properties carefully.
1. Domain:
The domain of a function represents all the possible input values (x-values) that the function can accept.
For the linear function [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex], there are no restrictions on the value of [tex]\( x \)[/tex]:
- There are no denominators that could cause division by zero.
- There are no square roots that require positive arguments.
- There are no logarithmic functions that require positive arguments.
Therefore, the function can accept all real numbers as input. This means the domain is:
[tex]\[
\text{domain: } (-\infty, \infty)
\][/tex]
2. Range:
The range of a function represents all the possible output values (y-values) that the function can produce.
For the linear function [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex], as [tex]\( x \)[/tex] takes all real values, the output [tex]\( f(x) \)[/tex] will also take all real values. This is because:
- The term [tex]\( \frac{3}{4} x \)[/tex] will cover all real values as [tex]\( x \)[/tex] covers all real numbers due to the continuous nature of linear functions.
- Adding a constant (+5) merely shifts the entire function up or down without restricting the range.
Therefore, the range is also all real numbers:
[tex]\[
\text{range: } (-\infty, \infty)
\][/tex]
Summary:
The correct domain and range for the function [tex]\( f(x) = \frac{3}{4} x + 5 \)[/tex] are:
[tex]\[
\text{domain: } (-\infty, \infty) \\
\text{range: } (-\infty, \infty)
\][/tex]
