In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).]
a)b)c)d)e)f)


$y = \sqrt{x - 4} + \sqrt{x + 4} - 4$

Verified step by step guidance

1. To find the x-intercepts of the graph, set the equation equal to zero because x-intercepts occur where the graph crosses the x-axis, meaning \(y = 0\). So, start with the equation: \$0 = \sqrt{\,x - 4\,} + \sqrt{\,x + 4\,} - 4$.
2. Isolate the square root terms on one side to simplify the equation. Add 4 to both sides to get: \$4 = \sqrt{\,x - 4\,} + \sqrt{\,x + 4\,}$.
3. To eliminate the square roots, consider squaring both sides of the equation. Remember, when squaring, use the formula \((a + b)^2 = a^2 + 2ab + b^2\). So, square both sides: \$4^2 = (\sqrt{\,x - 4\,} + \sqrt{\,x + 4\,})^2$.
4. Expand the right side using the formula: \$16 = (x - 4) + 2\sqrt{(x - 4)(x + 4)} + (x + 4)\(. Simplify the terms without the square root: \)16 = 2x + 2\sqrt{(x - 4)(x + 4)}$.
5. Isolate the square root term: \$16 - 2x = 2\sqrt{(x - 4)(x + 4)}\(. Then divide both sides by 2: \)\frac{16 - 2x}{2} = \sqrt{(x - 4)(x + 4)}\(. This simplifies to \)8 - x = \sqrt{x^2 - 16}\(. Next, square both sides again to eliminate the square root and solve for \)x$.

Key Concepts

Finding x-intercepts

X-intercepts are points where the graph crosses the x-axis, meaning the y-value is zero. To find them, set y = 0 in the equation and solve for x. This helps identify key points that characterize the graph's shape and position.

Domain of Radical Functions

For functions involving square roots, the expression inside the root must be non-negative to yield real values. Determining the domain involves solving inequalities like x - 4 ≥ 0 and x + 4 ≥ 0, which restricts the possible x-values and affects where the graph exists.

Graph Matching Using Intercepts

Matching equations to graphs often relies on key features like intercepts and domain restrictions. By finding x-intercepts and understanding the domain, you can compare these points and intervals to the given graphs to identify the correct match.