Question

Find the inverse of ℎ⁡(𝑥)=𝑥−4/3

. Graph the function and its inverse on a separate sheet of paper. If necessary, restrict the domain of the inverse so that it is a function.

Answer 1

Interpreting h(x) = (x − 4)/3 (the common meaning), find the inverse:

1. Write y = (x − 4)/3.
2. Swap x and y: x = (y − 4)/3.
3. Solve for y: 3x = y − 4, so y = 3x + 4.

Thus h^{-1}(x) = 3x + 4. Domain and range: both are (−∞, ∞) so no restriction is needed.

Graphing note: h(x) is a line with slope 1/3 and y-intercept −4/3; h^{-1}(x) is its reflection across y = x, a line with slope 3 and y-intercept 4. For example h(4)=0, so h^{-1}(0)=4.

If you instead meant h(x) = x − 4/3, then the inverse is h^{-1}(x) = x + 4/3 (also defined for all real x).