Math Problem Statement
The function f is defined as follows:
f(x) =
-3x + 5 if x < 1
4x - 2 if x >= 1
(a) Find the domain of the function.
(b) Locate any intercepts.
(c) Graph the function.
(d) Based on the graph, find the range.
Solution
To solve this piecewise-defined function problem, let's go through each part of the question step-by-step.
Given Function
The function is defined as:
\begin{cases} -3x + 5 & \text{if } x < 1 \\ 4x - 2 & \text{if } x \geq 1 \end{cases}$$ ### Part (a): Domain of the Function The domain of a function consists of all possible values of $$x$$ for which $$f(x)$$ is defined. - For $$x < 1$$, $$f(x) = -3x + 5$$ is defined. - For $$x \geq 1$$, $$f(x) = 4x - 2$$ is defined. Since both parts of the function cover all real numbers without any restrictions, the domain of $$f$$ is all real numbers. In interval notation, this is: $$(-\infty, \infty)$$ ### Part (b): Intercepts To find the intercepts, we look for points where $$f(x) = 0$$ (x-intercepts) and where $$x = 0$$ (y-intercepts). #### X-Intercept(s) 1. **For $$x < 1$$:** Set $$-3x + 5 = 0$$. $$-3x + 5 = 0 \Rightarrow x = \frac{5}{3}$$ Since $$\frac{5}{3} \approx 1.67$$ (greater than 1), this solution does not fall in the interval $$x < 1$$, so there is no x-intercept for this part of the function. 2. **For $$x \geq 1$$:** Set $$4x - 2 = 0$$. $$4x - 2 = 0 \Rightarrow x = \frac{1}{2}$$ Since $$\frac{1}{2} < 1$$, this does not fall in the interval $$x \geq 1$$, so there is no x-intercept for this part of the function either. There are no x-intercepts for $$f(x)$$. #### Y-Intercept To find the y-intercept, set $$x = 0$$. - For $$x = 0$$, we use the part of the function where $$x < 1$$, which is $$f(x) = -3x + 5$$. $$f(0) = -3(0) + 5 = 5$$ So the y-intercept is $$(0, 5)$$. ### Part (c): Graph the Function To graph $$f(x)$$, plot each piece on its respective interval. 1. **For $$x < 1$$:** $$f(x) = -3x + 5$$ - This is a linear function with slope $$-3$$ and y-intercept $$5$$. - For $$x \to 1^-$$, $$f(x) \to -3(1) + 5 = 2$$. - So, in this region, we have a line passing through $$(0, 5)$$ and approaching $$(1, 2)$$ from the left. 2. **For $$x \geq 1$$:** $$f(x) = 4x - 2$$ - This is a linear function with slope $$4$$ and y-intercept $$-2$$. - For $$x = 1$$, $$f(x) = 4(1) - 2 = 2$$. - So, in this region, we have a line starting at $$(1, 2)$$ and continuing with a positive slope. The graph will show a continuous function with no gaps at $$x = 1$$. ### Part (d): Range of the Function The range is the set of all possible values of $$f(x)$$. - As $$x \to -\infty$$, $$f(x) = -3x + 5$$ will tend towards $$+\infty$$. - As $$x \to +\infty$$, $$f(x) = 4x - 2$$ will also tend towards $$+\infty$$. - The minimum value of $$f(x)$$ occurs at $$f(1) = 2$$. Therefore, the range of $$f(x)$$ is: $$[2, \infty)$$ --- Would you like more details on any part of this solution? Here are some additional questions to deepen understanding: 1. How would the domain change if one of the intervals was restricted to a smaller subset? 2. How would the function's continuity be affected if there was a gap or jump at $$x = 1$$? 3. Can you identify other functions where the range is similarly constrained by a minimum or maximum value? 4. What happens to the intercepts if the function was transformed with a vertical shift? 5. How would the graph look if the slope of each piece was reversed? **Tip:** When working with piecewise functions, always evaluate boundary points carefully to ensure continuity or identify any potential jumps.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Domain and range
Intercepts
Graphing linear functions
Formulas
Domain: all x-values where the function is defined
Range: all possible values of f(x)
Intercepts: f(x) = 0 for x-intercept, x = 0 for y-intercept
Theorems
-
Suitable Grade Level
High School (Grades 9-12)
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