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Chapter 2: Problem 32

If \(f(x)=x \log x\) and \(g(x)=10^{x},\) then \(g(f(2))=\) (A) 24 (B) 17 (C) 4 (D) 2 (E) 0.6

Short Answer

Expert verified
Therefore, the result of \(g(f(2))\) is 4, so the correct answer is (C) 4

Step by step solution

01

Evaluate f(2)

Since \(f(x) = x \log x\), plugging \(x = 2\) into \(f\), we get \(f(2) = 2 \log 2\). The base of the logarithm is 10. Using the property \(\log_{b} b = 1\), \(\log 2\) in base 10 equals approximately 0.301, therefore, \(f(2) = 2 \times 0.301 = 0.602\).
02

Substitute f(2) into g(x)

Now that we have calculated \(f(2)\) to be 0.602, we can substitute that value into \(g(x) = 10^{x}\) to find \(g(f(2))\). Therefore, \(g(f(2)) = g(0.602) = 10^{0.602}\).
03

Calculate \(10^{0.602}\)

Recall that any number to the power of 0 is 1 and to the power of 1 is the number itself. As 0.602 is between 0 and 1, \(10^{0.602}\) should be between 1 and 10. The exact value of \(10^{0.602}\) can be found using a calculator, which is approximately 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Composition
Function composition involves combining two or more functions to produce a new function. When you see notation like \(g(f(x))\), this means you need to find the output of \(f(x)\) first and then use this result as the input for \(g(x)\).

This process can be understood through a simple series of steps:
  • Evaluate the inside function first. Here, it was \(f(x)\), which needed computing at a specific point \(x = 2\).
  • Once you have \(f(2)\), use this result as the input for the next function \(g(x)\), so you compute \(g(f(2))\).
  • Implement the output of \(f\) successfully into \(g\) to find the final result.
Composing functions allows us to build complex operations that can be managed step by step, often simplifying complex problems by breaking them into smaller, more manageable parts.
Exponential Functions
Exponential functions are functions of the form \(a^x\) where \(a\) is a constant, and \(x\) is the variable. These functions grow rapidly, especially as the input to the function increases.

In the exercise, \(g(x) = 10^x\) is an exponential function where 10 is the base and \(x\) determines the exponent. Here are some key points about exponential functions:
  • When \(x = 0\), \(a^x = 1\), because any non-zero number raised to the power of 0 is always 1.
  • If \(0 < x < 1\), the result will be between 1 and the base of the function, which is between 1 and 10 for \(10^x\).
  • As \(x\) increases above 1, the function鈥檚 outputs grow rapidly, reflecting exponential growth.
Exponential functions are powerful tools in modeling real-world phenomena where change happens at a constant multiplicative rate, like compound interest, population growth, and more.
SAT Math Preparation
For SAT math preparation, understanding the properties of logarithmic and exponential functions is crucial, as these topics often appear in different forms in the exam.

Key preparation tips include:
  • Make sure you practice simplifying expressions with logarithms and exponents to work more efficiently.
  • Keep common logarithms (like base 10) and their properties at the forefront of your prep. Recognize that \(\log_{10} b = x\) means \(10^x = b\).
  • Understand function composition as it often requires you to apply multiple skills in tandem, like plugging into functions and then interpreting results.
Practicing these concepts can greatly help in boosting your confidence and efficiency during the test, as familiarity with these types of functions can save invaluable time on exam day.

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Most popular questions from this chapter

If \(f(x)=\frac{x-7}{x^{2}-49}\) , for what value(s) of \(x\) does the graph of \(y\) \(=f(x)\) have a vertical asymptote? (A) \(-7\) (B) 0 (C) \(-7,0,7\) (D) \(-7,7\) (E) \(-7,7\) (E) 7

If \(\sin A=\frac{3}{5}, 90^{\circ} < A < 180^{\circ}, \cos B=\frac{1}{3},\) and \(270^{\circ} < B < \) \(360^{\circ},\) the value of \(\sin (A+B)\) is (A) \(-0.83\) (B) \(-0.55\) (C) \(-0.33\) (D) 0.73 (E) 0.95

Let \(f(x)=\sqrt{x^{3}-4 x}\) and \(g(x)=3 x .\) The sum of all values of \(x\) for which \(f(x)=g(x)\) is (A) \(-8.5\) (B) 0 (C) 8 (D) 9 (E) 9.4

If \(4.05^{p}=5.25^{q},\) what is the value of \(\frac{p}{q} ?\) (A) \(-0.11\) (B) 0.11 (C) 1.19 (D) 1.30 (E) 1.67

\(\left(-\frac{1}{16}\right)^{2 / 3}=\) (A) \(-0.25\) (B) \(-0.16\) (C) 0.16 (D) 6.35 (E) The value is not a real number.
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