Given that $$ f(x)=\frac{x}{2}+5 $$ a) Find $$ f(2) $$ b) Find $$ f^{-1}(x) $$ c) Find $$ f^{-1}(5) $$

Question

UpStudy AI · Accepted Answer

Let's solve each part of the problem step by step.

Given the function:
$$ f(x) = \frac{x}{2} + 5 $$

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### **a) Find $$ f(2) $$**

To find $$ f(2) $$, substitute $$ x = 2 $$ into the function:

$$
f(2) = \frac{2}{2} + 5 = 1 + 5 = 6
$$

**Answer:** $$ f(2) = 6 $$

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### **b) Find $$ f^{-1}(x) $$**

The inverse function $$ f^{-1}(x) $$ reverses the roles of $$ x $$ and $$ y $$ in the original function. Here's how to find it:

1. **Start with the original function:**
   $$
   y = \frac{x}{2} + 5
   $$

2. **Swap $$ x $$ and $$ y $$:**
   $$
   x = \frac{y}{2} + 5
   $$

3. **Solve for $$ y $$:**
   $$
   x - 5 = \frac{y}{2}
   $$
   $$
   y = 2(x - 5)
   $$
   $$
   y = 2x - 10
   $$

4. **Write the inverse function:**
   $$
   f^{-1}(x) = 2x - 10
   $$

**Answer:** $$ f^{-1}(x) = 2x - 10 $$

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### **c) Find $$ f^{-1}(5) $$**

Now, use the inverse function found in part (b) to find $$ f^{-1}(5) $$:

$$
f^{-1}(5) = 2(5) - 10 = 10 - 10 = 0
$$

**Answer:** $$ f^{-1}(5) = 0 $$

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### **Summary of Answers:**

a) $$ f(2) = 6 $$

b) $$ f^{-1}(x) = 2x - 10 $$

c) $$ f^{-1}(5) = 0 $$
a) $$ f(2) = 6 $$ b) $$ f^{-1}(x) = 2x - 10 $$ c) $$ f^{-1}(5) = 0 $$

Given that \( f(x)=\frac{x}{2}+5 \) a) Find \( f(2) \) b) Find \( f^{-1}(x) \) c) Find \( f^{-1}(5) \)