The function `f: R->R` defined by `f(x)=6^x+6^(|x|)` is (a) one-one and onto (b) many one and onto (c) one-one and into (d) many one and into

Explanation

To determine the nature of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 6^x + 6^{|x|} \), we need to analyze its properties, specifically whether it is one-one (injective) or many-one (not injective), and whether it is onto (surjective) or into (not surjective).

### Step 1: Analyze the function for \( x < 0 \) and \( x \geq 0 \)

1. **For \( x < 0 \)**:
- Here, \( |x| = -x \).
- Thus, \( f(x) = 6^x + 6^{-x} \).

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