Evaluate f(g(x)) for x ≤ 0 using Absolute Value Functions

Math Problem Statement

Let f(x)=|x|+xf(x)=|x|+x and g(x)=|x|−xg(x)=|x|−x for x∈Rx∈R then find the value of f(g(x))f(g(x)) for x≤0.x≤0.

We are tasked with finding $f(g(x))$ for $x \leq 0$ , where:

$f(x) = |x| + x, \quad g(x) = |x| - x.$

The absolute value function $|x|$ is defined as:

\begin{cases} x, & \text{if } x \geq 0, \\ -x, & \text{if } x < 0. \end{cases}$$ ### Step 2: Simplify $$g(x)$$ for $$x \leq 0$$ For $$x \leq 0$$, $$|x| = -x$$. Substituting this into $$g(x)$$, we get: $$g(x) = |x| - x = (-x) - x = -2x.$$ ### Step 3: Substitute $$g(x)$$ into $$f(x)$$ Now, substitute $$g(x) = -2x$$ into $$f(x)$$: $$f(g(x)) = f(-2x).$$ For $$f(x) = |x| + x$$, we compute $$f(-2x)$$ by considering $$-2x$$ for $$x \leq 0$$. Since $$x \leq 0$$, $$-2x \geq 0$$. Thus, $$| -2x | = -2x$$. Substituting: $$f(-2x) = |-2x| + (-2x) = (-2x) + (-2x) = -4x.$$ ### Step 4: Value of $$f(g(x))$$ for $$x \leq 0$$ Since $$x \leq 0$$, $$-4x = 0$$ because the term cancels out to $$0$$. Thus: $$f(g(x)) = 0.$$ ### Final Answer: $$\boxed{0}.$$ Would you like me to clarify any steps or explore related concepts? Here are some questions for further exploration: 1. What happens to $$f(g(x))$$ for $$x > 0$$? 2. How does the behavior of $$f(x)$$ and $$g(x)$$ change across the domain of $$x$$? 3. What is the graphical representation of $$f(g(x))$$ for $$x \leq 0$$? 4. How can piecewise functions like $$f(x)$$ and $$g(x)$$ be useful in modeling real-world situations? 5. Can you derive the general form of $$f(g(x))$$ for all $$x \in \mathbb{R}$$? **Tip:** When working with absolute value functions, always split the domain based on the sign of the input. This simplifies analysis significantly!

Drop file here or Click Here to upload

Absolute Value Functions
Function Composition
Piecewise Functions

|x| = x if x ≥ 0, and |x| = -x if x < 0
f(x) = |x| + x
g(x) = |x| - x
f(g(x)) = f(-2x) = -4x for x ≤ 0

Grades 9-11

Evaluate f(g(-3)) with Function Composition and Absolute Value

Composition of Functions: f(x) = 3|x + 1| - 4 and g(x) = -2x^2 - 5

Solving Function Composition Problems Involving Linear and Absolute Value Functions

Evaluating Absolute Value Functions with f(x) = -2 |x - 5|

Evaluating Absolute Value Function f(x) = |x|/x at Specific Values