Let's break down and solve each question:

1. Find the x- and y-intercepts of $f(x) = 2|x - 5| - 6$ algebraically.

Finding the y-intercept:

To find the y-intercept, set $x = 0$: $f(0) = 2|0 - 5| - 6 = 2 \times 5 - 6 = 10 - 6 = 4$ So, the y-intercept is at $(0, 4)$.

Finding the x-intercepts:

To find the x-intercepts, set $f(x) = 0$ and solve for $x$: $2|x - 5| - 6 = 0$ First, isolate the absolute value term: $2|x - 5| = 6$ $|x - 5| = 3$ This absolute value equation leads to two cases:

1. $x - 5 = 3$ ⟹ $x = 8$
2. $x - 5 = -3$ ⟹ $x = 2$

So, the x-intercepts are at $(8, 0)$ and $(2, 0)$.

2. Graph this function on Desmos, then find the domain and range in two different notations.

Graph:

The function $f(x) = 2|x - 5| - 6$ represents a V-shaped graph opening upwards, with the vertex located at $(5, -6)$.

Domain and Range:

• Domain: The function is defined for all real values of $x$, so the domain is:

  • Interval notation: $(-\infty, \infty)$
  • Set notation: $\{ x \in \mathbb{R} \}$

• Range: The lowest point on the graph is the vertex at $y = -6$. Since the function opens upwards, $f(x)$ takes on all values from $-6$ to $\infty$:

  • Interval notation: $[-6, \infty)$
  • Set notation: $\{ y \in \mathbb{R} \mid y \geq -6 \}$

Would you like more details on any part of this solution?

Related Questions:

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