Question

Given $f (x) = 4(x-3)^3+6,$, classify each statement about $f^-1(x)$ as true or false.

The point of symmetry on $f^-1(x)$ is (6,3)

The domain of $f^-1(x)$ is $(-\infty} , \infty})$

$f^-1(x)$ is not a function

The range of $f^-1(x)$ is $(- \infty},6)$

Answer 1

Answer:

• False
• True
• False
• False

Step-by-step explanation:

You want to know which of various statements about f(x) = 4(x- 3)³ +6 are true.

a) Symmetry

The cubic function is symmetric about its point of inflection. The translation by (3, 6) moves that to coordinates (3, 6). The point of symmetry is not (6, 3). (False)

b) Domain

The domain of any polynomial function is (-∞, ∞). True.

c) Inverse function

The slope of f(x) never changes sign, so the inverse relation is a function. (False)

d) Range of inverse

The range of the inverse relation is the same as the domain of the function. It is (-∞, ∞). (False)

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Additional comment

The function is graphed in red; the inverse function is graphed in green. The orange dashed line is the line of reflection between the function and its inverse: y = x. Both have domain and range of (-∞, ∞).

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