Directions: Read the following information and answer the two items that follow: Let f(x) = x 2, g(x) = tan x and h(x) = In x.
For \(x = \frac{{\sqrt \pi }}{2}\) , what is the value of [ho(gof)](x)?
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Let's break down this problem involving composite functions. We are given three functions: \(f(x) = x^2\), \(g(x) = \tan x\), and \(h(x) = \ln x\). We need to find the value of the composite function \([ho(gof)](x)\) when \(x = \frac{{\sqrt \pi }}{2}\).
The expression \([ho(gof)](x)\) represents applying the functions in a specific order: first \(f\), then \(g\) to the result of \(f\), and finally \(h\) to the result of \(g\). In other words, \([ho(gof)](x) = h(g(f(x)))\).
Evaluating Composite Functions Step-by-Step
To find \([ho(gof)]\left(\frac{{\sqrt \pi }}{2}\right)\), we will evaluate the functions from the inside out.
Step 1: Evaluate \(f(x)\) at \(x = \frac{{\sqrt \pi }}{2}\)
The first function applied is \(f(x) = x^2\). We need to find \(f\left(\frac{{\sqrt \pi }}{2}\right)\).
\(f\left(\frac{{\sqrt \pi }}{2}\right) = \left(\frac{{\sqrt \pi }}{2}\right)^2\)
Squaring the term gives:
\(\left(\frac{{\sqrt \pi }}{2}\right)^2 = \frac{(\sqrt \pi)^2}{2^2} = \frac{\pi}{4}\)
So, \(f\left(\frac{{\sqrt \pi }}{2}\right) = \frac{\pi}{4}\).
Step 2: Evaluate \(g(\text{result from Step 1})\)
The next function applied is \(g(x) = \tan x\). We need to evaluate \(g\) at the result from Step 1, which is \(\frac{\pi}{4}\).
\(g\left(\frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right)\)
The value of \(\tan\left(\frac{\pi}{4}\right)\) is a standard trigonometric value:
\(\tan\left(\frac{\pi}{4}\right) = 1\)
So, \(g\left(f\left(\frac{{\sqrt \pi }}{2}\right)\right) = g\left(\frac{\pi}{4}\right) = 1\).
Step 3: Evaluate \(h(\text{result from Step 2})\)
The final function applied is \(h(x) = \ln x\). We need to evaluate \(h\) at the result from Step 2, which is 1.
\(h(1) = \ln(1)\)
The natural logarithm of 1 is:
\(\ln(1) = 0\)
So, \(h\left(g\left(f\left(\frac{{\sqrt \pi }}{2}\right)\right)\right) = h(1) = 0\).
Final Result for ho(gof)(x)
By evaluating the composite function step by step, we found that for \(x = \frac{{\sqrt \pi }}{2}\), the value of \([ho(gof)](x)\) is 0.
Thus, \([ho(gof)]\left(\frac{{\sqrt \pi }}{2}\right) = 0\).
| Step | Function | Input | Output |
|---|---|---|---|
| 1 | \(f(x) = x^2\) | \(x = \frac{{\sqrt \pi }}{2}\) | \(f\left(\frac{{\sqrt \pi }}{2}\right) = \frac{\pi}{4}\) |
| 2 | \(g(x) = \tan x\) | \(\frac{\pi}{4}\) | \(g\left(\frac{\pi}{4}\right) = 1\) |
| 3 | \(h(x) = \ln x\) | 1 | \(h(1) = 0\) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Composite Function | A function created by combining two or more functions, where the output of one function becomes the input of another. Notation like \(h(g(f(x)))\) or \((h \circ g \circ f)(x)\). |
| \(f(x) = x^2\) | Squaring function. Takes an input and squares it. |
| \(g(x) = \tan x\) | Tangent function. Takes an angle (in radians for this problem) and returns the ratio of sine to cosine. |
| \(h(x) = \ln x\) | Natural logarithm function. The inverse of the exponential function \(e^x\). Defined for \(x > 0\). |
| \(\tan\left(\frac{\pi}{4}\right)\) | Value of tangent at 45 degrees or \(\frac{\pi}{4}\) radians, which is 1. |
| \(\ln(1)\) | Value of the natural logarithm at 1, which is 0. |
Additional Information on Composite Functions and Domains
When working with composite functions like \([ho(gof)](x) = h(g(f(x)))\), it's important to consider the domains of the individual functions. The input \(x\) must be in the domain of \(f\). The output \(f(x)\) must be in the domain of \(g\). The output \(g(f(x))\) must be in the domain of \(h\).
- The domain of \(f(x) = x^2\) is all real numbers, \((-\infty, \infty)\). The input \(x = \frac{{\sqrt \pi }}{2}\) is in this domain.
- The domain of \(g(x) = \tan x\) is all real numbers except where \(\cos x = 0\), i.e., \(x \ne \frac{\pi}{2} + n\pi\) for any integer \(n\). The output of \(f(x)\) was \(\frac{\pi}{4}\), which is in the domain of \(\tan x\).
- The domain of \(h(x) = \ln x\) is all positive real numbers, \((0, \infty)\). The output of \(g(f(x))\) was 1, which is in the domain of \(\ln x\).
Since all intermediate values were valid inputs for the subsequent functions, the composite function is defined at \(x = \frac{{\sqrt \pi }}{2}\).
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