The function `f: R->R` defined by `f(x)=2^x+2^(|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 properties of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 2^x + 2^{|x|} \), we will analyze 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 \( x < 0 \)

For \( x < 0 \):
- The absolute value function gives \( |x| = -x \).
- Thus, we can rewrite \( f(x) \) as:
\[
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