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The homework seems to contain three distinct questions. The first asks for the value of 'm' given it's the middle term of an arithmetic progression: 17, m, 27. The second asks to find the inverse func

Mathematics

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3 parts
1
22
2

x+52\frac{x + 5}{2}

x

3
5

Step-by-step solution

9 steps

Here are the solutions to the questions:

Part 1· 2 steps

Arithmetic Mean

Given that $17, m, 27$ are consecutive terms of an arithmetic progression.

  1. In an arithmetic progression, the middle term is the arithmetic mean of its neighbors.

    m=17+272m = \frac{17 + 27}{2}

  2. Calculate the sum and divide.

    m=442m = \frac{44}{2} m=22m = 22

    The arithmetic mean, mm, is 22\boxed{22}.

Part 2

x+52\frac{x + 5}{2}

Inverse Function and Composite Function

A function ff is defined by f(x)=2x−5f(x) = 2x - 5.

  • Find f−1(x)f^{-1}(x)

  1. Let y=f(x)y = f(x), so y=2x−5y = 2x - 5.

    To find the inverse, swap xx and yy. x=2y−5 x = 2y - 5

  2. Solve for yy.

    $$
x + 5 = 2y
$$
$$
y = \frac{x + 5}{2}
$$
So, $f^{-1}(x) = \boxed{\frac{x + 5}{2}}$.
    • Find ff−1(x)f f^{-1}(x)

  3. Substitute f−1(x)f^{-1}(x) into f(x)f(x).

    $$
f f^{-1}(x) = f\left(\frac{x + 5}{2}\right)
$$

  4. Apply the function f(x)=2x−5f(x) = 2x - 5.

    $$
f\left(\frac{x + 5}{2}\right) = 2\left(\frac{x + 5}{2}\right) - 5
$$
$$
f f^{-1}(x) = (x + 5) - 5
$$
$$
f f^{-1}(x) = x
$$
So, $f f^{-1}(x) = \boxed{x}$.

Part 35

Value of xx if ff(x)=xf f(x) = x

Given f(x)=2x−5f(x) = 2x - 5.

  1. Find the composite function $f f(x)$.

    ff(x)=f(2x−5)f f(x) = f(2x - 5)

  2. Substitute 2x−52x - 5 into f(x)f(x).

    f(2x−5)=2(2x−5)−5f(2x - 5) = 2(2x - 5) - 5 ff(x)=4x−10−5f f(x) = 4x - 10 - 5 ff(x)=4x−15f f(x) = 4x - 15

  3. Set ff(x)=xf f(x) = x and solve for xx.

    4x−15=x4x - 15 = x 4x−x=154x - x = 15 3x=153x = 15 x=153x = \frac{15}{3} x=5x = 5

    The value of xx is 5\boxed{5}.

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