Question

In Exercises 7–10, determine from its graph if the function is one-to-one.f(x) = 3 - x, x \u003c 0      = 3, x ≥ 0

Accepted Answer

Understand the definition of a one-to-one function: a function is one-to-one if and only if each output value corresponds to exactly one input value. In other words, no horizontal line intersects the graph more than once. Analyze the given piecewise function: $$ f(x) = \begin{cases} 3 - x, & x \u003c 0 \ 3, & x \geq 0 \end{cases} . $$For $$ x \u003c 0 $$, the function is a line with slope $$-1$$, and for $$ x \geq 0 $$, the function is constant at 3. Consider the graph of the first part $$ 3 - x $$ for $$ x \u003c 0 . $$This is a decreasing linear function, so it is one-to-one on this interval. Look at the second part where $$ f(x) = 3 $$ for $$ x \geq 0 . $$This is a horizontal line, meaning all inputs $$ x \geq 0 $$ map to the same output 3, which violates the one-to-one condition on this interval. Check if any output value is repeated for different inputs across the two pieces. Since $$ f(0) = 3 $$ and also $$ f(x) = 3 $$ for all $$ x \geq 0 $$, and the first piece approaches values near 3 as $$ x 	o 0^- $$, the function is not one-to-one overall.

In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0

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Concetti chiave

One-to-One Function

Piecewise Functions

Graphical Test for One-to-One Functions

In Exercises 7–10, determine from its graph if the function is one-to-one.f(x) = 3 - x, x < 0 = 3, x ≥ 0