The function `f:R rarr R` defined by `f(x) = 6^x + 6^(|x|` is

Options

A. one - one and onto
B. many - one and onto
C. one - one and into
D. many - one and into

Answer

To analyze the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 6^x + 6^{|x|} \), we can break it down into two cases based on the value of \( x \).

### Step 1: Define the function for different cases
1. **Case 1**: When \( x \geq 0 \)
- Here, \( |x| = x \).
- Therefore, the function becomes:
\[ f(x) = 6^x + 6^x = 2 \cdot 6^x \]

2. **Case 2**: When \( x < 0 \)
- Here, \( |x| = -x \).
- Therefore, the function becomes:
\[ f(x) = 6^x + 6^{-x} = 6^x + \frac{1}{6^{-x}} = 6^x + \frac{1}{6^x} \]

### Step 2: Analyze the function for \( x \geq 0 \)
- For \( x \geq 0 \):
\[ f(x) = 2 \cdot 6^x \]
- This function is strictly increasing because the exponential function \( 6^x \) is increasing. Thus, \( f(x) \) is one-to-one in this interval.

### Step 3: Analyze the function for \( x < 0 \)
- For \( x < 0 \):
\[ f(x) = 6^x + \frac{1}{6^x} \]
- Let \( y = 6^x \), where \( y \) is positive and \( y < 1 \) since \( x < 0 \).
- The function can be rewritten as:
\[ f(x) = y + \frac{1}{y} \]
- The function \( g(y) = y + \frac{1}{y} \) is minimized when \( y = 1 \) (by AM-GM inequality), and its minimum value is \( 2 \). As \( y \) approaches \( 0 \), \( g(y) \) approaches \( \infty \). Thus, for \( x < 0 \), \( f(x) \) is also increasing and takes values from \( 2 \) to \( \infty \).

### Step 4: Combine the results
- At \( x = 0 \):
\[ f(0) = 6^0 + 6^0 = 1 + 1 = 2 \]
- Therefore, the function \( f(x) \) is continuous and takes the value \( 2 \) at \( x = 0 \) and increases to \( \infty \) in both directions.

### Step 5: Determine the range and co-domain
- The range of \( f(x) \) is \( [2, \infty) \).
- The co-domain is \( \mathbb{R} \).
- Since the range does not equal the co-domain, \( f(x) \) is not onto.

### Step 6: Determine if the function is one-to-one
- Since both segments of the function are strictly increasing, \( f(x) \) is one-to-one.

### Conclusion
The function \( f(x) = 6^x + 6^{|x|} \) is a one-to-one function with a range of \( [2, \infty) \) and is not onto \( \mathbb{R} \).