Mathematics

Mathematics

Problem

Consider the two functions defined on R\mathbb{R}: (i) g(x)=∣x∣g(x)=|x| (ii) h(x)=xh(x)=x Which one of the following statements about their ranges is correct? a) g(R)=[0,∞),  h(R)=[0,∞)g(\mathbb{R})=[0,\infty),\; h(\mathbb{R})=[0,\infty) b) g(R)=R,  h(R)=[0,∞)g(\mathbb{R})=\mathbb{R},\; h(\mathbb{R})=[0,\infty) c) g(R)=[0,∞),  h(R)=Rg(\mathbb{R})=[0,\infty),\; h(\mathbb{R})=\mathbb{R} d) g(R)=(0,∞),  h(R)=Rg(\mathbb{R})=(0,\infty),\; h(\mathbb{R})=\mathbb{R}
Intuitive explanation

The range of a function is the set of all possible output values it can actually produce. The key insight is to think about what yy-values each function can reach: for g(x)=∣x∣g(x) = |x|, absolute value always gives non-negative outputs, so it can never produce negative numbers; for h(x)=xh(x) = x, a linear function with slope 1 can produce any real number as you vary xx across all of R\mathbb{R}. Compare these two sets of achievable outputs to determine which statement about their ranges is correct.

Intuitive explanation

The range of a function is the set of all possible output values it can actually produce. The key insight is to think about what yy-values each function can reach: for g(x)=∣x∣g(x) = |x|, absolute value always gives non-negative outputs, so it can never produce negative numbers; for h(x)=xh(x) = x, a linear function with slope 1 can produce any real number as you vary xx across all of R\mathbb{R}. Compare these two sets of achievable outputs to determine which statement about their ranges is correct.

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