AP DSC School Assistant Math Model Test 1

Concept:A function is one-one (injective) if different inputs always give different outputs – that is, $f(x_1)=f(x_2)$ implies $x_1=x_2$. A function is onto (surjective) if every element of the codomain has at least one pre‑image in the domain.Explanation:Given $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^4$.1. One‑one check: Let $f(x_1)=f(x_2)$. Then $x_1^{4}=x_2^{4}$. Taking fourth roots gives $x_1 = \pm x_2$. This means two different inputs (e.g., $2$ and $-2$) give the same output $16$; hence $f$ is not one‑one.2. Onto check: For a real output $y$, we need $x^4 = y$. Solving, $x = \pm y^{1/4}$. If $y$ is negative (say $y=-1$), then $y^{1/4}$ is not a real number – there is no real $x$ such that $x^4$ is negative. Therefore not every real number has a pre‑image; $f$ is not onto.Thus the function is neither one‑one nor onto.Answer:Option B: f is neither one‑one nor onto.