Question
Find the inverse function. \[ f(x)=\frac{2 x+5}{4} \] \( f^{-1}(x)=\frac{[?] x+\square}{} \)
Ask by Johnston Burgess. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = \frac{4x - 5}{2} \).
Solution
To find the inverse function \( f^{-1}(x) \) of the given function \( f(x) = \frac{2x + 5}{4} \), follow these steps:
1. **Start with the original function:**
\[
y = \frac{2x + 5}{4}
\]
2. **Swap \( x \) and \( y \) to find the inverse:**
\[
x = \frac{2y + 5}{4}
\]
3. **Solve for \( y \):**
\[
4x = 2y + 5
\]
\[
2y = 4x - 5
\]
\[
y = \frac{4x - 5}{2}
\]
4. **Write the inverse function:**
\[
f^{-1}(x) = \frac{4x - 5}{2}
\]
So, the inverse function is:
\[
f^{-1}(x) = \frac{4x - 5}{2}
\]
**Answer:**
After simplifying, the inverse is (4 × x – 5) divided by 2. Thus,
f⁻¹(x) = (4x − 5)/2
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Simplify this solution Beyond the Answer
To find the inverse function, we start by replacing \( f(x) \) with \( y \): \[ y = \frac{2x + 5}{4} \] Now, we solve for \( x \) in terms of \( y \): 1. Multiply both sides by 4: \[ 4y = 2x + 5 \] 2. Subtract 5 from both sides: \[ 4y - 5 = 2x \] 3. Divide by 2: \[ x = \frac{4y - 5}{2} \] Now, we replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \frac{4x - 5}{2} \] So, in the requested format: \( f^{-1}(x)=\frac{4x - 5}{2} \)
