To determine the domain and range of the function [tex]\( f(x) = \sqrt{-9.3x + 70.5} \)[/tex], we need to analyze both the inside of the square root and the nature of the square root function itself. Here's how we can approach it step-by-step:
### Domain:
1. Identify the expression inside the square root:
The function [tex]\( f(x) = \sqrt{-9.3x + 70.5} \)[/tex] has [tex]\(-9.3x + 70.5\)[/tex] inside the square root.
2. Set the inside of the square root to be greater than or equal to zero:
For the square root to be defined, the expression inside must be non-negative:
[tex]\[
-9.3x + 70.5 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
70.5 \geq 9.3x
\][/tex]
Divide both sides by 9.3:
[tex]\[
x \leq \frac{70.5}{9.3}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{70.5}{9.3} = 7.58
\][/tex]
Hence, the values of [tex]\(x\)[/tex] must satisfy:
[tex]\[
x \leq 7.58
\][/tex]
5. Since there are no restrictions other than the upper bound:
The domain is:
[tex]\[
[0, 7.58]
\][/tex]
### Range:
1. Evaluate the function at the boundaries of the domain:
To find the range, we need to evaluate [tex]\( f(x) \)[/tex] at the minimum and maximum values of [tex]\( x \)[/tex] within the domain.
2. At the lower boundary [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \sqrt{-9.3 \cdot 0 + 70.5} = \sqrt{70.5}
\][/tex]
3. At the upper boundary [tex]\( x = 7.58 \)[/tex]:
[tex]\[
f(7.58) = \sqrt{-9.3 \cdot 7.58 + 70.5} = \sqrt{0} = 0
\][/tex]
4. Evaluate the numeric values:
After evaluating the function:
[tex]\[
\sqrt{70.5} \approx 8.396
\][/tex]
5. Determine the range:
Since the square root function outputs all non-negative values within the domain, the function [tex]\( f(x) \)[/tex] ranges from the value at [tex]\( x = 7.58 \)[/tex] up to the value at [tex]\( x = 0 \)[/tex]:
[tex]\[
[0, \sqrt{70.5}]
\][/tex]
Substituting the numerical value:
[tex]\[
[0, 8.396]
\][/tex]
### Interval Notation:
- Domain: The interval for [tex]\( x \)[/tex] can be written as:
[tex]\[
[0, 7.58]
\][/tex]
- Range: The interval for [tex]\( f(x) \)[/tex] can be written as:
[tex]\[
[0, 8.396]
\][/tex]
Thus, the domain and range of the function [tex]\( f(x) = \sqrt{-9.3x + 70.5} \)[/tex] are:
- Domain: [tex]\( [0, 7.58] \)[/tex]
- Range: [tex]\( [0, 0.0774596669240645] \)[/tex]
