Rewrite The Following Expression In Terms Of The Given Function

Rewrite the Expression in Terms of the Given Function: A Complete Guide

Mastering the art of rewriting expressions in terms of a given function is a foundational skill in algebra and calculus that unlocks deeper mathematical understanding. This process, often called function composition or algebraic substitution, involves taking a complex expression and systematically replacing its variables with another specified function. It’s not merely an academic exercise; it’s a critical tool for simplifying problems, analyzing rates of change, and modeling real-world systems. Whether you’re preparing for advanced mathematics exams or tackling engineering problems, the ability to fluently rewrite expressions transforms confusing equations into manageable, insightful forms. This guide will walk you through the conceptual framework, a reliable step-by-step methodology, and practical applications, ensuring you can approach any such problem with confidence.

Understanding the Core Concept: What Does "In Terms Of" Really Mean?

At its heart, the instruction "rewrite the expression in terms of the given function" means you are performing a substitution. You are given two pieces of information:

1. An original expression, typically involving a variable like x or t.
2. A definition of a new function, often named f(x), g(t), etc., which itself is expressed in terms of that same variable.

Your task is to take the original expression and replace every instance of the variable with the entire function definition. The result is a new, equivalent expression that now depends on the function itself as its core component, rather than the original variable. Think of it as translating a sentence from one language (the variable x) into another language (the function f(x)). The meaning remains identical, but the form changes to suit a new context. This skill is the algebraic bedrock for composite functions (like f(g(x))), changing variables of integration in calculus, and redefining physical quantities in terms of new parameters.

The Step-by-Step Methodology: A Systematic Approach

Success hinges on a disciplined, error-free process. Rushing or skipping steps is the primary source of mistakes. Follow this algorithm for any problem.

Step 1: Identify and Isolate

Clearly define your components. Write down:

• The original expression you need to rewrite (e.g., x² + 3x - 5).
• The given function definition (e.g., f(x) = 2x - 1).
• The variable of substitution (almost always x in these problems). Confirm that the variable in the original expression matches the variable in the function's definition.

Step 2: Solve the Function Definition for the Variable (If Necessary)

This is the most critical and often overlooked step. The function is given as f(x) = [expression in x]. To substitute into the original expression, you need the variable x isolated on one side of the equation.

• If the function is already solved for x (e.g., x = g(t)), you can proceed.
• If it is in the standard f(x) = ... form, you must algebraically rearrange it to solve for x.
  • Example: Given f(x) = 3x + 7, you solve for x: x = (f(x) - 7)/3. Now you have x expressed in terms of f(x).

Step 3: Perform the Substitution

Take the isolated expression for x from Step 2 and plug it directly into every single occurrence of x in the original expression. Use parentheses meticulously around the substituted phrase to avoid order-of-operations errors.

• Continuing Example: Original expression: E = x² - 4x. Substitution: x = (f - 7)/3.
• Substituted Expression: E = [ (f - 7)/3 ]² - 4[ (f - 7)/3 ].

Step 4: Simplify Relentlessly

Now, simplify the resulting expression. This involves:

1. Expanding parentheses (using (a/b)² = a²/b²).
2. Combining like terms.
3. Factoring where possible to achieve the most concise form.
4. Ensuring the final expression is written only in terms of the given function (e.g., f), with no remnants of the original variable x.

• Simplification: E = ( (f - 7)² / 9 ) - ( 4(f - 7) / 3 ) = ( (f² - 14f + 49) / 9 ) - ( (12f - 84) / 9 ) [Common denominator] = ( f² - 14f + 49 - 12f + 84 ) / 9 = ( f² - 26f + 133 ) / 9 Final Answer: (f² - 26f + 133)/9.

Scientific Explanation: Why This Process Works and Where It's Used

This procedure is fundamentally an application of the substitution property of equality. If a = b, then a can be replaced by b in any expression without changing the expression's value. Here, from f(x) = 3x + 7, we derive the true statement x = (f(x) - 7)/3. Therefore, substituting (f(x) - 7)/3 for x in any other expression containing x yields an equal expression.

The power of this technique manifests across STEM fields:

• Calculus (Change of Variables/u-Substitution): When integrating ∫ (2x) * (x² + 1)⁵ dx, we set u = x² + 1. Then du/dx = 2x, so du = 2x dx. We rewrite the entire integral in terms of u: ∫ u⁵ du. This simplification is only possible because we mastered expression rewriting.
• Physics and Engineering: A kinematic equation for position might be s(t) = s₀ + v₀t + ½at². If we are given that velocity v(t) = v₀ + at, we can solve for t (t = (v - v₀)/a) and rewrite the position equation entirely in terms of velocity v. This reveals the direct relationship s = s₀ + (v² - v₀²)/(2a), a form useful for analyzing motion without explicit time.
• Computer Science & Algorithms: Transforming a recursive formula T(n) = T(n-1) + n into a closed-form expression `T(n) = n(n

… + n into a closed‑form expression T(n) = n(n+1)/2. This derivation illustrates how rewriting a recurrence in terms of a simpler variable—here, the index n itself—eliminates the need for iterative computation and yields an immediate formula for algorithmic running time.

Additional Applications Across Disciplines

Economics:
Consider a demand function Q(p) = a - bp, where price p is expressed as a function of quantity sold: p = (a - Q)/b. Substituting this inverse relation into the revenue function R = p·Q gives R(Q) = Q·(a - Q)/b = (aQ - Q²)/b. The resulting quadratic makes it trivial to locate the profit‑maximizing quantity by setting the derivative to zero.

Statistics: In linear regression, the ordinary least squares estimator for the slope is β̂ = Σ(x_i - x̄)(y_i - ȳ) / Σ(x_i - x̄)². If we first center the predictor (u_i = x_i - x̄) and the response (v_i = y_i - ȳ), the formula simplifies to β̂ = Σ u_i v_i / Σ u_i². The substitution step—expressing each original variable as its deviation from the mean—clarifies that the slope depends only on covariances of centered data.

Control Theory:
A state‑space model might be written as ẋ = Ax + Bu, y = Cx + Du. When designing a state‑feedback controller u = -Kx + r, substituting the control law into the state equation yields ẋ = (A - BK)x + Br. The closed‑loop dynamics are now expressed solely in terms of the new matrix A_cl = A - BK, facilitating eigenvalue placement and stability analysis without repeatedly referencing the original input term.

Signal Processing:
The discrete Fourier transform (DFT) of a sequence x[n] is X[k] = Σ_{n=0}^{N-1} x[n] e^{-j2πkn/N}. If the signal is known to be a shifted version of a base sequence, x[n] = s[n - m], substituting gives X[k] = e^{-j2πkm/N} S[k], where S[k] is the DFT of s[n]. This property—called the shift theorem—emerges directly from the substitution step and underpins many filtering and modulation techniques.

Why Mastery Matters

The substitution technique is more than a mechanical algebraic trick; it is a mindset that encourages analysts to re‑express problems in the most convenient variables. By doing so, we:

1. Expose hidden structure (e.g., completing the square, recognizing a quadratic form).
2. Reduce computational burden (turning recurrences into closed forms, integrals into elementary antiderivatives). 3. Reveal invariants (showing that certain quantities depend only on combinations of original variables).
3. Facilitate communication across fields, since a common transformed expression often appears in disparate contexts (e.g., the quadratic (f² - 26f + 133)/9 mirrors energy expressions in physics and cost functions in economics).

Conclusion

From isolating a variable to plugging it back, simplifying, and interpreting the result, the four‑step substitution workflow provides a reliable pathway to rewrite any expression in terms of a chosen function or variable. Its utility spans calculus, physics, engineering, computer science, economics, statistics, control theory, and signal processing—demonstrating that a solid grasp of this fundamental procedure empowers problem‑solvers to transform complex relationships into tractable, insightful forms. Whenever faced with an entangled equation, remember: solve for the desired term, substitute meticulously, simplify relentlessly, and let the new expression reveal the underlying simplicity.