if y = f(x) such that (2^x +4^y =4) find range

Given the equation \( 2^x + 4^y = 4 \), where \( y = f(x) \), we need to find the range of the function \( y = f(x) \).

### Step 1: Express the equation in a more manageable form
First, we can rewrite \( 4^y \) as \( (2^2)^y = 2^{2y} \). So the equation becomes:
\[
2^x + 2^{2y} = 4
\]
Since \( 4 = 2^2 \), we have:
\[
2^x + 2^{2y} = 2^2
\]

### Step 2: Analyze possible values for \( 2^x \) and \( 2^{2y} \)
For \( 2^x + 2^{2y} = 2^2 = 4 \), both \( 2^x \) and \( 2^{2y} \) must be non-negative powers of 2. Also, because both terms must add up to 4, let's explore possible cases:

1. **Case 1:** \( 2^x = 1 \) and \( 2^{2y} = 3 \)  
   This case doesn't work because \( 3 \) is not a power of \( 2 \).

2. **Case 2:** \( 2^x = 2 \) and \( 2^{2y} = 2 \)  
   This implies \( x = 1 \) and \( 2y = 1 \), so \( y = \frac{1}{2} \).

3. **Case 3:** \( 2^x = 4 \) and \( 2^{2y} = 0 \)  
   This case is invalid since \( 2^{2y} \) cannot be \( 0 \) (as \( 2^{2y} \) must be positive).

Thus, from Case 2, we find that the only value that works is \( y = \frac{1}{2} \) when \( x = 1 \).

### Step 3: Determine the range
From the analysis above, \( y \) can only take the value \( \frac{1}{2} \). Thus, the range of the function \( y = f(x) \) is:
\[
\boxed{\left\{ \frac{1}{2} \right\}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How can we find the domain for \( y = f(x) \) based on the given equation?
2. What if the equation were \( 2^x + 4^y = 8 \)? How would that change the range?
3. How does the equation \( 2^x + 2^{2y} = 4 \) relate to logarithmic functions?
4. Can the solution method be generalized for similar equations involving exponents?
5. What would happen to the range if we changed the base of the exponent from 2 to another number?

**Tip:** When dealing with equations involving exponents, always check whether the terms can be simplified by expressing them with a common base. This can greatly simplify the problem.

Discover how to determine the range of the function y = f(x) from the equation 2^x + 4^y = 4. Explore step-by-step solutions and understand the relationship between x and y in exponential equations. Suitable for high school students studying functions and exponential equations.

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