Given f(x) = { 9-x², x<2 √x + 7, 2≤ x < 10, give the values of the following: [x-4] , x ≥ 10

(a) f(2)

(b) f(12.5)

(c) f(-3)

(d) f(5)

(e) f(1.5)

To solve this problem, we'll break it into steps using the piecewise function \( f(x) \) provided:

1. \( f(x) = 9 - x^2 \) for \( x < 2 \)
2. \( f(x) = \sqrt{x} + 7 \) for \( 2 \leq x < 10 \)
3. \( f(x) = [x - 4] \) for \( x \geq 10 \), where \( [x-4] \) likely refers to the greatest integer function (also known as the floor function) \( \lfloor x-4 \rfloor \).

Now let's find each of the given values:

### (a) \( f(2) \)
Since \( 2 \leq x < 10 \):
\[
f(2) = \sqrt{2} + 7
\]
\[
f(2) \approx 1.414 + 7 = 8.414
\]

### (b) \( f(12.5) \)
Since \( x \geq 10 \):
\[
f(12.5) = \lfloor 12.5 - 4 \rfloor = \lfloor 8.5 \rfloor = 8
\]

### (c) \( f(-3) \)
Since \( x < 2 \):
\[
f(-3) = 9 - (-3)^2 = 9 - 9 = 0
\]

### (d) \( f(5) \)
Since \( 2 \leq x < 10 \):
\[
f(5) = \sqrt{5} + 7
\]
\[
f(5) \approx 2.236 + 7 = 9.236
\]

### (e) \( f(1.5) \)
Since \( x < 2 \):
\[
f(1.5) = 9 - (1.5)^2 = 9 - 2.25 = 6.75
\]

### Summary of Results
(a) \( f(2) \approx 8.414 \)  
(b) \( f(12.5) = 8 \)  
(c) \( f(-3) = 0 \)  
(d) \( f(5) \approx 9.236 \)  
(e) \( f(1.5) = 6.75 \)

Would you like more details on any of these steps, or do you have any questions?

### Related Questions
1. What is the general approach for evaluating a piecewise function at a given point?
2. How do you interpret the greatest integer function (\( \lfloor x \rfloor \)) in mathematics?
3. Can you explain how to calculate square roots for numbers that are not perfect squares?
4. How do you determine the domain of a piecewise function?
5. What is the significance of evaluating functions at specific points?

### Tip
When dealing with piecewise functions, always check which interval the input value falls into before applying the function rule.

Given f(x) = { 9-x², x<2 √x + 7, 2≤ x < 10, give the values of the following: [x-4] , x ≥ 10 (a) f(2) (b) f(12.5) (c) f(-3) (d) f(5) (e) f(1.5)

Explore how to solve a piecewise function with conditions on x values. This problem involves using the square root function, quadratic equations, and the floor function for specific intervals. Includes detailed steps for evaluating f(x) at different points, like f(2), f(12.5), and more. Suitable for high school students in Grades 9-11.

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