Determine whether each pair of functions graphed are inverses.

Verified step by step guidance

1. Identify the two functions graphed. The blue line and the orange line are the two functions in question.
2. Determine the equation of the line y = x, which is the green dashed line. This line is the line of symmetry for inverse functions.
3. Check if the blue line and the orange line are reflections of each other across the line y = x. This means that for every point (a, b) on the blue line, there should be a corresponding point (b, a) on the orange line.
4. Observe the points where the blue and orange lines intersect the line y = x. If they intersect at the same points, they are likely inverses.
5. Verify by selecting a few points on the blue line and checking if their coordinates are swapped on the orange line. If this holds true for multiple points, the functions are inverses.

Key Concepts

Inverse Functions

Inverse functions are functions that 'reverse' the effect of the original function. If a function f takes an input x and produces an output y, then its inverse f⁻¹ takes y as input and returns x. For two functions to be inverses, they must satisfy the condition f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in their domains.

Graphical Representation of Inverses

Graphically, two functions are inverses if their graphs are symmetric with respect to the line y = x. This means that if a point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f⁻¹. This symmetry visually confirms that the functions undo each other.

Function Composition

Function composition involves combining two functions to form a new function. For two functions f and g, the composition f(g(x)) applies g first and then f to the result. To verify if two functions are inverses, checking their compositions (f(f⁻¹(x)) and f⁻¹(f(x))) is essential, as both should yield the identity function, which is simply x.