Answered

What are the domain and range of the function [tex]f(x) = \frac{3}{4}x + 5[/tex]?

A. Domain: [tex](-\infty, \infty)[/tex], Range: [tex](-\infty, \infty)[/tex]

B. Domain: [tex]\left(\frac{3}{4}, \infty\right)[/tex], Range: [tex](-\infty, 5)[/tex]

C. Domain: [tex]\left(-\infty, \frac{3}{4}\right)[/tex], Range: [tex](5, \infty)[/tex]

D. Domain: [tex]\left(\frac{3}{4}, \infty\right)[/tex], Range: [tex](5, \infty)[/tex]



Answer :

GinnyAnswer
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]