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f(x) = 22 + 7 and h(2x) = * - 7

2
Write simplified expressions for fh(x)) and h(f(x)) in terms of x.
f(h(x)) =
f
h(f(x)) =
极
Are functions f and h inverses?

1 answer

Let's start by simplifying the functions provided and writing the compositions \( f(h(x)) \) and \( h(f(x)) \).

Firstly, we have the functions defined as:

  • \( f(x) = 22 + 7 \)
  • \( h(2x) = * - 7 \)

Here, there's something odd about the expressions you've provided for \( f(x) \) and \( h(2x) \). It seems that the definition of \( f(x) \) is actually a constant function because it simplifies to \( f(x) = 29 \) for any input \( x \).

However, the expression for \( h(2x) \) is unclear as it contains a "*". I will assume we can consider \( h(2x) = 2x - 7 \) for the sake of this example, as it seems to be a reasonable linear function. So we will use:

  • \( f(x) = 29 \)
  • \( h(x) = 2x - 7 \) where \( h(x) \) is expressed in terms of \( x \).

Now, let's find \( f(h(x)) \) and \( h(f(x)) \):

  1. Calculate \( f(h(x)) \): \[ f(h(x)) = f(2x - 7) = 29 \quad \text{(since \( f(x) \) is constant)} \]

  2. Calculate \( h(f(x)) \): \[ h(f(x)) = h(29) = 2(29) - 7 = 58 - 7 = 51 \]

Thus, we have: \[ f(h(x)) = 29 \] \[ h(f(x)) = 51 \]

Are functions \( f \) and \( h \) inverses?

Two functions \( f \) and \( h \) are inverses of each other if:

  1. \( f(h(x)) = x \) for all \( x \) in the domain of \( h \)
  2. \( h(f(x)) = x \) for all \( x \) in the domain of \( f \)

From our calculations:

  • \( f(h(x)) = 29 \) (not equal to \( x \))
  • \( h(f(x)) = 51 \) (not equal to \( x \))

Since neither condition holds, we conclude that \( f \) and \( h \) are not inverses.